AP CALCULUS BC
Final Notes
Trigonometric Formulas
9.
sin   cos   1
1  tan 2   sec 2 
1  cot 2   csc 2 
sin(  )   sin 
cos(  )  cos
tan(  )   tan 
sin( A  B)  sin A cos B  sin B cos A
sin( A  B)  sin A cos B  sin B cos A
cos( A  B)  cos A cos B  sin A sin B
10.
cos( A  B)  cos A cos B  sin A sin B
1.
2.
3.
4.
5.
6.
7.
8.
2
2
13.
14.
15.
16.
17.
18.
sin 
1

cos  cot 
cos 
1
cot  

sin  tan 
1
sec  
cos 
1
csc  
sin 
1
cos 2   1  cos 2 
2
tan  
sin 2  
sin 2  2 sin  cos
2
2
2
2
12. cos 2  cos   sin   2 cos   1  1  2 sin 
11.
1
1  cos 2 
2
Differentiation Formulas
1.
2.
3.
4.
5.
6.
7.
8.
9.
d n
( x )  nx n 1
dx
d
( fg )  fg   gf 
dx
d f
gf   f g 
( )
dx g
g2
d
f ( g ( x))  f ( g ( x)) g ( x)
dx
d
(sin x)  cos x
dx
d
(cos x)   sin x
dx
d
(tan x)  sec 2 x
dx
d
(cot x)   csc 2 x
dx
d
(sec x)  sec x tan x
dx
10.
11.
12.
13.
14.
15.
16.
17.
d
(csc x)   csc x cot x
dx
d x
(e )  e x
dx
d x
(a )  a x ln a
dx
d
1
(ln x) 
dx
x
d
1
( Arc sin x) 
dx
1 x2
d
1
( Arc tan x) 
dx
1 x2
d
1
( Arc sec x) 
dx
| x | x2 1
dy dy du


Chain Rule
dx dx dx
Integration Formulas
1.
 a dx  ax  C
2.
n
 x dx 
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
x n 1
 C , n  1
n 1
1
 x dx  ln x  C
 e dx  e  C
x
x
ax
C
ln a
 ln x dx  x ln x  x  C
x
 a dx 
 sin x dx   cos x  C
 cos x dx  sin x  C
 tan x dx  ln sec x  C or  ln cos x  C
 cot x dx  ln sin x  C
 sec x dx  ln sec x  tan x  C
 csc x dx  ln csc x  cot x  C   ln csc x  cot x  C
 sec x d x  tan x  C
 sec x tan x dx  sec x  C
 csc x dx   cot x  C
 csc x cot x dx   csc x  C
 tan x dx  tan x  x  C
2
2
2
dx
1
x
 Arc tan    C
2
a
x
a
dx
 x
 Arc sin    C
19. 
2
2
a
a x
x
dx
1
1
a
20. 
 Arc sec  C  Arc cos  C
a
a
x
x x2  a2 a
18.
a
2
1.
Formulas and Theorems
Limits and Continuity:
A function y  f (x) is continuous at x  a if
i).
f(a) exists
ii).
lim f ( x) exists
iii).
xa
lim  f (a)
xa
Otherwise, f is discontinuous at x = a.
The limit lim f  x  exists if and only if both corresponding one-sided limits exist and are equal –
x a
that is,
lim f  x   L  lim f  x   L  lim f  x 
x a
2.
3.
x a
Note: The period of the function
The amplitude is
4.
x a
Even and Odd Functions
1.
A function y  f (x) is even if f ( x)  f ( x) for every x in the function’s domain.
Every even function is symmetric about the y-axis.
2.
A function y  f (x) is odd if f ( x)   f ( x) for every x in the function’s
domain.
Every odd function is symmetric about the origin.
Periodicity
A function f (x ) is periodic with period p ( p  0) if f ( x  p )  f ( x) for every value of
x.
y  A sin( Bx  C ) or y  A cos( Bx  C ) is
A . The period of y  tan x is  .
Intermediate-Value Theorem
A function y  f (x) that is continuous on a closed interval
a, b takes on every value
f (a) and f (b) .
Note: If f is continuous on a, b and f (a ) and f (b ) differ in sign, then the equation
f ( x)  0 has at least one solution in the open interval (a, b) .
Limits of Rational Functions as x  
f ( x)
lim
 0 if the degree of f ( x)  the degree of g ( x)
i).
x   g ( x)
x 2  2x
Example: lim
0
x   x3  3
f ( x)
ii).
is infinite if the degrees of f ( x)  the degree of g ( x)
lim
x   g ( x )
x3  2x
Example: lim

x   x2  8
f ( x)
iii).
is finite if the degree of f ( x)  the degree of g ( x)
lim
x   g ( x )
between
5.
2
.
B
Example:
2 x 2  3x  2
2
lim

5
x   10 x  5 x 2
6.
Horizontal and Vertical Asymptotes
1.
A line y  b is a horizontal asymptote of the graph
2.
7.
lim f ( x)  b or lim f ( x)  b .
x
x  
A line x  a is a vertical asymptote of the graph y  f (x) if either
lim f ( x)   or lim   .
x  a
x  a-
 0 0  and x1, y1  are points on the graph of
Average and Instantaneous Rate of Change
i).
Average Rate of Change: If x , y
ii).
8.
y  f (x) , then the average rate of change of y with respect to x over the interval
x0 , x1  is f ( x1 )  f ( x0 )  y1  y0  y .
x1  x0
x1  x0 x
Instantaneous Rate of Change: If x0 , y 0  is a point on the graph of y  f (x) , then
the instantaneous rate of change of y with respect to x at x 0 is f ( x0 ) .
Definition of Derivative
f ( x)  lim
h0
9.
ii).
11.
a, b and differentiable on a, b such that f (a) 
is at least one number c in the open interval a, b such that f (c)  0 .
Mean Value Theorem
If f is continuous on
a, b such that
a, b.
f (b) , then there
a, b and differentiable on a, b , then there is at least one number
a, b,
then
f (x) has both a maximum and minimum
Absolute Mins and Maxs: To find the maximum and minimum values of a function
locate
1.
the points where f (x ) is zero or where f (x ) fails to exist.
2.
c
f (b)  f (a )
 f (c) .
ba
Extreme-Value Theorem
If f is continuous on a closed interval
on
13.
n
 1
lim 1    e
n   n 
1
 nn
lim 1    e
n  0 1 
Rolle’s Theorem
If f is continuous on
in
12.
f ( x  h)  f ( x )
f  x   f a 
or f '  a   lim
x a
x a
h
The latter definition of the derivative is the instantaneous rate of change of f  x  with respect to
x at x = a.
Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of
the function at that point.
The Number e as a limit
i).
10.
y  f (x) if either
the end points, if any, on the domain of
y  f (x) ,
f (x) .
Plug those values into f (x ) to see which gives you the max and which gives you this
min values (the x-value is where that value occurs)
Note: These are the only candidates for the value of x where f (x ) may have a maximum or a
minimum.
3.
14.
Increasing and Decreasing: Let
f be differentiable for a  x  b and continuous for a
a  x  b,
f ( x)  0 for every x in a, b , then f is increasing on a, b .
2.
If f ( x )  0 for every x in a, b , then f is decreasing on a, b .
Concavity: Suppose that f (x) exists on the interval a, b
1.
If f ( x )  0 in a, b , then f is concave upward in a, b .
2.
If f ( x )  0 in a, b , then f is concave downward in a, b .
To locate the points of inflection of y  f (x) , find the points where f ( x )  0 or where
f (x) fails to exist. These are the only candidates where f (x) may have a point of inflection.
Then test these points to make sure that f ( x )  0 on one side and f ( x )  0 on the other.
1.
15.
16a.
16b.
If
If a function is differentiable at point x  a , it is continuous at that point. The converse is false,
in other words, continuity does not imply differentiability.
Local Linearity and Linear Approximations
The linear approximation to f (x ) near x 
x0 is given by y  f ( x0 )  f ( x0 )( x  x0 ) for
x sufficiently close to x 0 .
To estimate the slope of a graph at a point – just draw a tangent line to the graph at that point.
Another way is (by using a graphing calculator) to “zoom in” around the point in question until the
graph “looks” straight. This method almost always works. If we “zoom in” and the graph looks
straight at a point, say  a, f  a   , then the function is locally linear at that point.
The graph of y  x has a sharp corner at x = 0. This corner cannot be smoothed out by
“zooming in” repeatedly. Consequently, the derivative of x does not exist at x = 0, hence, is not
locally linear at x = 0.
17.
Dominance and Comparison of Rates of Change
Logarithm functions grow slower than any power function  x n  .
Among power functions, those with higher powers grow faster than those with lower powers.
All power functions grow slower than any exponential function  a x , a  1 .
Among exponential functions, those with larger bases grow faster than those with smaller bases.
We say, that as x   :
1. f  x  grows faster than g  x  if lim
x 
f x
g

  or if lim
x 
gx
f x
 0.
If f  x  grows faster than g  x  as x   , then g  x  grows slower than f  x  as
x.
f x
2. f  x  and g  x  grow at the same rate as x   if lim
 L  0 (L is finite
x  g  x 
and nonzero).
For example,
ex

x  x 3
x4

2. x 4 grows faster than ln x as x   since lim
x  ln x
1. e x grows faster than x 3 as x   since lim
x 2  2x
1
x 
x2
To find some of these limits as x   , you may use the graphing calculator. Make sure that an
appropriate viewing window is used.
3. x 2  2x grows at the same rate as x 2 as x   since lim
18.
L’Hôpital’s Rule
f ( x)
f ( x)
0

or
lim
is of the form
, and if lim
exists, then
0

x  a g ( x)
x  a g ( x)
f ( x)
f ( x)
lim
 lim
.
x  a g ( x) x  a g ( x)
If
19.
Inverse function
1.
If f and
g are two functions such that f ( g ( x))  x for every x in the domain of
g and g ( f ( x))  x for every x in the domain of f , then f and g are inverse
3.
functions of each other.
A function f has an inverse if and only if no horizontal line intersects its graph more
than once.
If f is strictly either increasing or decreasing in an interval, then f has an inverse.
4.
If
2.
f is differentiable at every point on an interval I , and f ( x)  0 on I , then
g  f 1 ( x) is differentiable at every point of the interior of the interval f (I ) and if
1
the point  a, b  is on f  x  , then the point  b, a  is on g  f ( x) ; furthermore
1
.
f 'a
x
Properties of y  e
g ' b 
20.
1.
2.
3.
4.
5.
6.
7.
8.
21.
x
The exponential function y  e is the inverse function of
The domain is the set of all real numbers,    x   .
The range is the set of all positive numbers, y  0 .

2.
3.
4.
5.
6.

d x
d f  x
(e )  e x and
e
 f '  x  e f  x
dx
dx
x
x
x  x2
e 1 e 2  e 1
y  e x is continuous, increasing, and concave up for all x .
lim e x   and
lim e x  0 .
x  
x  
e ln x  x , for x  0; ln( e x )  x for all x .
Properties of
1.
y  ln x .
y  ln x
y  ln x is the set of all positive numbers, x  0 .
The range of y  ln x is the set of all real numbers,    y   .
y  ln x is continuous and increasing everywhere on its domain.
ln ab  ln a  ln b .
a
ln    ln a  ln b .
b
ln a r  r ln a .
The domain of
8.
y  ln x  0 if 0  x  1.
lim ln x   and lim ln x   .
x  
x  0
9.
log
10.
f ' x
d
d
1
ln f  x   
ln  x   
and


dx
f  x
dx
x
7.
22.
a
x
ln x
ln a
Trapezoidal Rule
If a function f is continuous on the closed interval
partitioned into
a, b where a, b has been equally
x0 , x1 , x1, x2 , ...xn  1, xn 
n subintervals
, each length

b

ba
, then
n
ba
f ( x )  2 f ( x )  2 f ( x )  ...  2 f ( x
)  f ( x ) , which is
0
1
2
n 1
n
2n
1
equivalent to  Leftsum  Rightsum 
2
 f ( x) dx 
a
23a.
Definition of Definite Integral as the Limit of a Sum
Suppose that a function f (x ) is continuous on the closed interval
a, b .
Divide the interval into
ba
. Choose one number in each subinterval, in other
n
words, x in the first, x in the second, …, x in the k th ,…, and x in the n th . Then
k
n
2
1
b
n
lim
 f ( x )x   f ( x) dx  F (b)  F (a) .
k
n  k 1
a
n equal subintervals, of length x 
23b.
Properties of the Definite Integral
Let f (x ) and g (x ) be continuous on
b
b
a, b.
 c  f ( x) dx  ca f ( x) dx for any constant c .
a
a
ii).  f ( x) dx  0
a
b
a
iii).  f (x) dx    f (x) dx
a
b
b
c
b
iv).  f ( x) dx   f ( x) dx   f ( x) dx , where f is continuous on an interval
a
a
c
containing the numbers a, b, and c .
a
v). If f (x ) is an odd function, then  f ( x) dx  0
a
i).
vi). If
vii). If
f (x) is an even function, then
f ( x)  0 on a, b , then
a
a
a
0
 f ( x) dx  2  f ( x) dx
b
 f ( x) dx  0
a
viii). If
g ( x)  f ( x) on a, b , then
b
b
 g ( x) dx  a f ( x) dx
a
24.
Fundamental Theorem of Calculus:
b

f ( x) dx  F (b)  F (a), where F ( x)  f ( x), or
a
Second Fundamental Theorem of Calculus (Steve’s Theorem):
25.
g x 
x
d
f (t) dt  f (x)
dx a
26.
b
d
f ( x) dx  f ( x) .
dx a
or
d
f (t) dt  g '  x  f (g  x )  h '  x  f  h  x  
dx hx 
Velocity, Speed, and Acceleration
1. The velocity of an object tells how fast it is going and in which direction. Velocity is an
instantaneous rate of change. If velocity is positive (graphically above the “x”-axis), then the
object is moving away from its point of origin. If velocity is negative (graphically below the
“x”-axis), then the object is moving back towards its point of origin. If velocity is 0
(graphically the point(s) where it hits the “x”-axis), then the object is not moving at that time.
2. The speed of an object is the absolute value of the velocity, v  t  . It tells how fast it is going
disregarding its direction.
The speed of a particle increases (speeds up) when the velocity and acceleration have the same
signs. The speed decreases (slows down) when the velocity and acceleration have opposite
signs.
3. The acceleration is the instantaneous rate of change of velocity – it is the derivative of the
velocity – that is, a  t   v '  t  . Negative acceleration (deceleration) means that the velocity is
decreasing (i.e. the velocity graph would be going down at that time), and vice-versa for
acceleration increasing. The acceleration gives the rate at which the velocity is changing.
Therefore, if x is the displacement of a moving object and t is time, then:
dx
i) velocity = v  t   x '  t  
dt
ii) acceleration = a  t   x ''  t   v '  t  
dv d 2 x

dt dt 2
iii) v  t    a  t  dt
iv) x  t    v  t  dt
Note: The average velocity of a particle over the time interval from t 0 to another time t, is
Average Velocity =
b
at time t or
Change in position s  t   s  t 0 

, where s  t  is the position of the particle
Length of time
t  t0
1
v  t  dt if given the velocity function.
b  a a
b
1
f ( x) dx .
b  a a
f (x) on a, b is
27.
The average value of
28.
Area Between Curves
If f and g are continuous functions such that
b
curves is
  f ( x)  g ( x) dx
or
a
29.
Integration By “Parts”
If u  f (x) and v 
f ( x)  g ( x) on a, b , then area between the
b
d
a
c
 top  bottom dx or  [right  left ]dy .
g (x ) and if f (x) and g (x ) are continuous, then
 u dx  uv   v du .
Note: The goal of the procedure is to choose
u and dv so that
 v du is easier to solve
than the original problem.
Suggestion:
When “choosing” u , remember L.I.A.T.E, where L is the logarithmic function, I is an
inverse trigonometric function, A is an algebraic function, T is a trigonometric function, and E is
the exponential function. Just choose u as the first expression in L.I.A.T.E (and dv will be the
 x ln x dx , choose u  ln x
dv  x dx . When integrating  xe x dx , choose u  x ,
remaining part of the integrand). For example, when integrating
since L comes first in L.I.A.T.E, and
x is an algebraic function, and A comes before E in L.I.A.T.E, and dv  e x dx . One
more example, when integrating  x Arc tan( x ) dx , let u  Arc tan( x) , since I comes before
since
A in L.I.A.T.E, and
30.
dv  x dx .
Volume of Solids of Revolution (rectangles drawn perpendicular to the axis of revolution)
 Revolving around a horizontal line (y=# or x-axis) where a  x  b :
Axis of Revolution below the region being revolved:
b
V     top function  a.r.   bottom function  a.r. dx
2
2
a
Axis of Revolution above the region being revolved:
b
V     bottom function  a.r.   top function  a.r. dx
2
2
a

c yd:
Axis of Revolution is left of the region being revolved:
Revolving around a vertical line (x=# or y-axis) where
d
V     right function  a.r.   left function  a.r. dy
2
2
c
Axis of Revolution is right of the region being revolved:
d
V     left function  a.r.   right function  a.r. dy
2
c
2
30b.
Volume of Solids with Known Cross Sections
b
1.
For cross sections of area
A(x ) , taken perpendicular to the x-axis, volume =
a A( x) dx .
Cross-sections {if only one function is used then just use that function, if it is between two functions use
top-bottom} mostly all the same only varying by a constant, with the only exception being the rectangular
cross-sections:
 Square cross-sections:
b
V    top function  bottom function  dx
2
a

Equilateral cross-sections:
b
3
2
V
 top function  bottom function  dx

4 a

Isosceles Right Triangle cross-sections (hypotenuse in the xy plane):
b
V

1
2
 top function  bottom function  dx

4a
Isosceles Right Triangle cross-sections (leg in the xy plane):
b
V

1
2
 top function  bottom function  dx

2a
Semi-circular cross-sections:
V

b
 top function  bottom function 
8
2
dx
a

Rectangular cross-sections (height function or value must be given or articulated
somehow – notice no “square” on the {top – bottom} part):
b
V    top function  bottom function  (height function / value)dx
a

Circular cross-sections with the diameter in the xy plane:
V

b
 top function  bottom function 
4
2
dx
a

Square cross-sections with the diagonal in the xy plane:
b
1
2
V    top function  bottom function  dx
2a
b
2.
For cross sections of area
A( y ) , taken perpendicular to the y-axis, volume =
 A( y) dy .
a
30c.
Shell Method (used if function is in terms of x and revolving around a vertical line) where
a xb:
b
V  2  r ( x)h( x)dx
a
r ( x)  x
if a.r. is y-axis (x  0)
r ( x)  ( x  a.r.) if a.r. is to the left of the region
r ( x)  (a.r.  x) if a.r. is to the right of the region
h( x)  f ( x) if only revolving with one function
h( x)   top  bottom  if revolving the region between two functions
31.
Solving Differential Equations: Graphically and Numerically
Slope Fields
dy
 f  x, y  gives the slope of the
At every point  x, y  a differential equation of the form
dx
member of the family of solutions that contains that point. A slope field is a graphical
representation of this family of curves. At each point in the plane, a short segment is drawn whose
slope is equal to the value of the derivative at that point. These segments are tangent to the
solution’s graph at the point.
The slope field allows you to sketch the graph of the solution curve even though you do not have
its equation. This is done by starting at any point (usually the point given by the initial condition),
and moving from one point to the next in the direction indicated by the segments of the slope field.
Some calculators have built in operations for drawing slope fields; for calculators without this
feature there are programs available for drawing them.
Euler’s Method
Euler’s Method is a way of approximating points on the solution of a differential equation
dy
 f  x, y  . The calculation uses the tangent line approximation to move from one point to the
dx
next. That is, starting with the given point  x1 , y1  – the initial condition, the point
x
1
 x, y1  f '  x1 , y1  x  approximates a nearby point on the solution graph. This
aproximation may then be used as the starting point to calculate a third point and so on. The
accuracy of the method decreases with large values of x . The error increases as each successive
point is used to find the next. Calculator programs are available for doing this calculation.
32.
Logistics
1. Rate is jointly proportional to its size and the difference between a fixed positive number (L)
and its size.
dy
dy
 y
 ky  M  y  which yields
 ky 1   OR
dt
dt
 L
y
33.
L
1  Ce  kt
through separation of variables
2.
lim y  L ; L = carrying capacity (Maximum); horizontal asymptote
3.
y-coordinate of inflection point is
t 
Definition of Arc Length
L
, i.e. when it is growing the fastest (or max rate).
2
y  f (x) represents a smooth curve on the interval a, b , then the arc
b
2
length of f between a and b is given by s   1   f ( x) dx .
a
If the function given by
34.
Improper Integral
b
 f ( x) dx is an improper integral if
a
35.
1. f becomes infinite at one or more points of the interval of integration, or
2. one or both of the limits of integration is infinite, or
3. both (1) and (2) hold.
Parametric Form of the Derivative
If a smooth curve C is given by the parametric equations x  f ( x) and y 
slope of the curve C at
( x, y ) is
Note: The second derivative,
g (t ) , then the
dy dy dx dx

 ,
 0.
dx dt dt dt
d 2 y d  dy  d  dy  dx

    .
dx 2 dx  dx  dt  dx  dt
36.
Arc Length in Parametric Form
If a smooth curve C is given by
37.
Polar Coordinates
1. Cartesian vs. Polar Coordinates. The polar coordinates
x  f (t ) and y  g (t ) and these functions have continuous
first derivatives with respect to t for a  t  b , and if the point P ( x, y ) traces the curve
exactly once as t moves from t  a to t  b , then the length of the curve is given by
2
2
b
b
 dx 
 dy 
2
2
s        dt    f (t )  g (t ) dt .
 dt 
a  dt 
a
2
2
speed  f (t)  g(t)
coordinates
(r , ) are related to the Cartesian
( x, y ) as follows:
x  r cos  and y  r sin 
y
tan   and x 2  y 2  r 2
x
2. To find the points of intersection of two polar curves, find (r , ) satisfying the first equation
for which some points (r ,  2n ) or (r ,    2n ) satisfy the second equation.
Check separately to see if the origin lies on both curves, i.e. if r can be 0. Sketch the curves.
3. Area in Polar Coordinates: If f is continuous and nonnegative on the interval  ,   , then
the area of the region bounded by the graph of r  f ( ) between the radial lines   
and    is given by

4.

1
1
2
A    f ( ) d   r 2 d
2
2
Derivative of Polar function: Given r  f   , to find the derivative, use parametric
equations.
x  r cos   f   cos 
and
y  r sin   f   sin  .
5.
38.
dy
f    cos   f '    sin 
dy
Then
 d 
dx dx
 f    sin   f '    cos 
d

2
 dr 
2
Arc Length in Polar Form: s   r  
 d
 d 

Sequences and Series
1. If a sequence an has a limit
 
L , that is, lim a n  L , then the sequence is said to
n
converge to L . If there is no limit, the series diverges. If the sequence an  converges,
then its limit is unique. Keep in mind that
ln n
lim
 0;
n n





1 


lim x n   1;
n
lim n n  1;
n
xn
lim
 0 . These limits
n n!
are useful and arise frequently.
 1

a
n
diverges; the geometric series  ar converges to
1 r
n1 n
n0
if r  1 and diverges if r  1 and a  0 .

2.
The harmonic series
3.
The p-series
4.
Limit Comparison Test: Let

1
p converges if p  1 and diverges if p  1 .
n
n1



a
and
 n
 bn be a series of nonnegative terms, with
n1
n1
b
an  0 for all sufficiently large n , and suppose that lim n  c  0 . Then the two
n an
series either both converge or both diverge.

5.
Alternating Series: Let
i)
ii)
iii)
 an
n1
be a series such that
the series is alternating
an1  an for all n , and
lim a  0
n n
Then the series converges.
Alternating Series Remainder: The remainder
R N is less than (or equal to) the first
neglected term
R N  a N1
6.
The n-th Term Test for Divergence: If lim a n  0 , then the series diverges.
n 
Note that the converse is false, that is, if lim a n  0 , the series may or may not converge.
n 
7.
 an is absolutely convergent if the series  an converges. If  an
converges, but  an does not converge, then the series is conditionally convergent. Keep


in mind that if  a n converges, then  a n converges.
A series
n1
8.
Comparison Test: If
n1
0  an  bn for all sufficiently large n , and

then
9.


a
a
converges.
If
diverges,
then
 n
 n
 bn diverges.
n1
n1
n1

 bn converges,
n1
f (x) is a positive, continuous, and decreasing function on 1,  and let


an  f (n) . Then the series  a n will converge if the improper integral  f ( x) dx
n1
1


converges. If the improper integral  f ( x) dx diverges, then the infinite series  a n
n1
1
Integral Test: If
diverges.
10. Ratio Test: Let
 an
i)
If
a
lim n1  1 , then the series converges absolutely.
n a n
ii)
If
a
lim n1  1 , then the series is divergent.
n a n
iii)
If
a
lim n1  1 , then the test is inconclusive (and another test
n a n
be a series with nonzero terms.
must be used).
11. Power Series: A power series is a series of the form

 cn x n  c0  c1x  c2 x 2  ...  cn x n  ...
or
n0

 cn ( x  a) n  c0  c1( x  a)  c2 ( x  a) 2  ...  cn ( x  a) n  ... in which the
n0
center a and the coefficients c , c , c ,..., cn ,... are constants. The set of all numbers x
0 1 2
for which the power series converges is called the interval of convergence.
f be a function with derivatives of all orders throughout some intervale
containing a as an interior point. Then the Taylor series generated by f at a is
( n)
 f (k ) (a)
f (a)
f
(a)
k
2

 k! ( x  a)  f (a)  f (a)( x  a)  2! ( x  a)  ...  n! ( x  a) n  ...
k 0
12. Taylor Series: Let
The remaining terms after the term containing the
remainder to Taylor’s Theorem:
nth derivative can be expressed as a
x
n (n)
1
(n1)
f ( x)  f ( a )   f
(a)( x  a) n  Rn ( x) where Rn ( x)   ( x  t ) n f
(t ) dt
n
!
a
1
Lagrange’s form of the remainder:
(n1)
(c) | ( x  a) |n1
, where a  c  x .
(n  1)!
The series will converge for all values of x for which the remainder approaches zero as
| f  x   Pn  x  || Rn x |
f
x.
13. Frequently Used Series and their Interval of Convergence

1
 1  x  x 2  ...  x n  ...   x n , x  1
1 x
n0
 xn
x2
xn
x
e  1 x 
 ... 
 ...  
, x 
2!
n!
n0 n!
sin x  x 
 (1) n x 2n1
x3 x5
x 2n1

 ...  (1) n
 ...  
, x 
3!
5!
(2n  1)!
n0 (2n  1)!
cos x  1 
 (1) x 2n
x2 x4
x 2n

 ...  (1) n
 ...  
, x 
2!
4!
(2n)!
(
2
n
)!
n0