Calculus
Homework # 4
Homework # 5
8.6 # 14: (4 pts, p. 603) Find a power series representation for the function and determine
the radius of convergence.
f (x) = x2 tan−1 x3
To do this, use the formula
on p. 601-602 of the textbook for tan−1 x. This formula was
∫ dx
1
generated from the integral 1+x2 and the series for 1+x
2 . The series needed is:
2
f (x) = x tan
−1
3
2
x =x
∞
∑
(−1)n
n=0
=
∞
∑
(−1)n
n=0
(x3 )2n+1
2n + 1
x6n+5
2n + 1
8.7 # 6: (3 pts, p. 616) Find the Maclaurin series for f (x) using the definition of a Maclaurin
series. [Assume that f has a power series expansion. Do not show that Rn (x) → 0.] Also, find
the associated radius of convergence.
f (x) = ln(1 + x)
The values of the derivatives of the function f at x = 0 are:
f (x) = ln(1 + x)
1
f (1) (x) =
1+x
1
(2)
f (x) = −
(1 + x)2
2
f (3) (x) =
(1 + x)3
6
f (4) (x) = −
(1 + x)4
(n − 1)!
f (n) (x) = (−1)n−1
(1 + x)n
f (0) = ln(1 + 0) = 0
1
f (1) (0) =
=1
1+0
1
f (2) (0) = −
= −1
(1 + 0)2
2
f (3) (0) =
=2
(1 + 0)3
6
f (4) (0) = −
= −6
(1 + 0)4
(n − 1)!
f (n) (0) = (−1)n−1
= (−1)n−1 (n − 1)!
(1 + 0)n
Then the Maclaurin series is (the first term is equal to zero):
f (x) =
∞
∑
f (n) (0)
n=0
n!
xn =
=
∞
∑
(−1)n−1 (n − 1)!
n=1
∞
∑
n=1
n!
(−1)n+1 (n
− 1)!
n!
xn =
xn =
∞
∑
(−1)n−1
n=1
∞
∑
n=1
1
n
xn
(−1)n+1 n
x
n
Same value...
Calculus
Homework # 4
The radius of convergence can be found from the limit of the ratio of two successive terms:
(−1)n
n+1 x
lim n+1n−1 < 1
n→∞ (−1)
n
x n
nx <1
lim n→∞ n + 1 |x| < 1
−1 < x < 1
So the radius of convergence is 1 (nothing is determined here about whether the series converges
for x = −1 or x = 1).
8.8 # 5: (3 pts, p. 625) Find the Taylor polynomial T3 (x) for the function f at the number a.
Graph f and T3 on the same screen.
f (x) = cos x ,
a=
π
2
To find the Taylor polynomial T3 requires the first 3 derivatives of the function f . These
are:
f ( π2 ) = 0
f (1) (x) = − sin x
f (1) ( π2 ) = −1
f (2) (x) = − cos x
f (2) ( π2 ) = 0
f (3) (x) = sin x
f (3) ( π2 ) = 1
Now the Taylor polynomial T3 is:
T3 (x) = −(x − π2 ) + 13 (x − π2 )3
(Not required) Here is a graph of this Taylor polynomial (blue) and cos x (red) which shows
how close the Taylor polynomial is to the value of cos x when x is near π2 .
2
Calculus
Homework # 4
1
y
0.5
0
-0.5
-1
0
0.5
1
1.5
x
3
2
2.5
3