Sequence/Series Review
The following is intended to give you an overview of the material/concepts that we have
covered in chapter 9. These topics will be covered on the third midterm.
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Sequences
1. Formally a sequence is a function f : N → R defined by f (n) = an . Informally, we can
think of a sequence as a list of real numbers, one for each natural number.
2. Given a sequence {an } we can ask whether or not an converges or diverges, namely,
whether lim an < ∞ (i.e. converges) or lim an = ∞ or does not exist (i.e. diverges).
n→∞
n→∞
All of the rules we have developed regarding limits in general apply to taking limits of
sequences, in particular L’Hopital’s rule can be applied in appropriate situations.
3. Let an , bn be convergent sequences and k a real number, then we have the following
properties
(a) lim k = k
n→∞
(b) lim kan = k lim an
n→∞
n→∞
(c) lim (an + bn ) = lim an + lim bn
n→∞
n→∞
n→∞
(d) lim (an − bn ) = lim an − lim bn
n→∞
n→∞
n→∞
(e) lim (an bn ) = ( lim an )( lim bn )
n→∞
n→∞
n→∞
lim an
an
= n→∞ , provided lim bn 6= 0.
n→∞
n→∞ bn
lim bn
(f) lim
n→∞
4. If |r| < 1 then lim r n = 0.
n→∞
5. (Squeeze Theorem) Let an , bn , and cn be sequences with an ≤ bn ≤ cn . If lim an = lim cn = L,
n→∞
then lim bn = L.
n→∞
1
n→∞
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Introduction to Series
1. An infinite series is an infinite sum of the form
∞
X
an = a1 + a2 + . . .
n=1
2. Let S =
∞
X
an be an infinite series. The n-th partial sum of S is Sn =
n=1
n
X
ak = a1 + a2 + . . . + an .
k=1
We say that the series S converges if the sequence of partial sums {Sn } converges, otherwise we say that the series diverges.
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Series
1. A series of the form
series converges to
∞
X
ar n−1 is called a geometric series. If |r| < 1, then this type of
n=1
a
. (Notice
1−r
∞
X
1
we should perform partial fractions first on
n(n + 2)
n=1
so that the series becomes a collapsing series. We can determine the sum of
2. Given a series of the form
1
n(n+2)
that a is simply the first term in the series).
such a series by using partial sums.
∞
X
1
3. A series of the form
is called a p-series and converges for p > 1 and diverges for
np
n=1
p ≤ 1.
4. A series of the form
∞
X
(−1)n+1 an , where an ≥ 0 is called an alternating series.
n=1
5. Given the series
∞
X
un we say that this series converges absolutely if the series
n=1
converges. If
conditionally.
∞
X
|un | diverges but
n=1
6. A series of the form
∞
X
un converges, then we say that
n=1
∞
X
∞
X
∞
X
|un |
n=1
un converges
n=1
an xn is called a power series in the variable x centered around
n=0
0, or simply just a power series.
7. A series of the form
∞
X
an (x − a)n is called a power series in the variable x centered
n=0
around a, or simply just a power series.
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8. Given a power series of the form
one and only one of the following:
∞
X
an (x − a)n the convergence set for this series is
n=0
(a) All real numbers, i.e. R; the radius of convergence is R = ∞
(b) The interval (a − R, a + R) (possibly with endpoints), where R is called the radius
of convergence of the series.
(c) Only the point x = a; the radius of convergence is R = 0.
9. If S(x) =
∞
X
an xn is a power series on some interval I, for x in the interior of I we
n=0
can differentiate and integrate S(x) term-by-term to obtain a new series, i.e. for x in
the interior of I we have
′
(a) S (x) =
(b)
Z
∞
X
an nxn−1
n=1
x
S(t)dt =
0
∞ Z
X
n=0
x
an tn dt
0
10. Using differentiation and integration of power series we can obtain formulas for ln(1+x),
tan−1 (x), and ex .
11. Suppose that f satisfies f (x) = c0 + c1 (x − a) + c2 (x − a)2 + . . . for all x in some interval
around a, then cn =
f (n) (a)
.
n!
12. (Taylor’s Formula) Let f be a function which has (n + 1) derivatives for each x in some
interval I containing a. Then, for each x in I we have
f (x) = f (a) + f ′ (a)(x − a) +
Where Rn (x) =
f (n+1) (c)
(x
(n+1)!
f ′′ (a)
f (n) (a)
(x − a)2 + . . . +
(x − a)n + Rn (x)
2!
n!
− a)n+1 for some point c in between x and a.
13. (Taylor’s Theorem) Assume that f has derivatives of all orders in some interval of a,
then the Taylor series of f is
f (x) =
∞
X
f (n) (a)
n=0
if and only if lim Rn (x) = 0.
n→∞
3
n!
(x − a)n
14. A Maclaurin series is a Taylor’s series with a = 0.
15. Important Maclaurin series:
1
= 1 + x + x2 + x3 + . . .
1−x
4
−1<x<1
ln(1 + x) = x −
x2 x3 x4
+
−
+ ...
2
3
4
−1 <x≤1
tan−1 (x) = x −
x3 x5 x7
+
−
+ ...
3
5
7
−1 ≤x≤1
ex = 1 + x +
x2 x3 x4
+
+
+ ...
2!
3!
4!
sin(x) = x −
x3 x5 x7
+
−
+ ...
3!
5!
7!
cos(x) = 1 −
x2 x4 x6
+
−
+ ...
2!
4!
6!
sinh(x) = x +
x3 x5 x7
+
+
+ ...
3!
5!
7!
cosh(x) = 1 +
x2 x4 x6
+
+
+ ...
2!
4!
6!
Theorems for Series
1. (n-th term test for divergence) If the series
alently, if lim an 6= 0, then
n→∞
∞
X
∞
X
n=1
an diverges.
n=1
4
an converges, then lim an = 0. Equivn→∞
2. (Integral Test) If 0 < an+1 < an , set f (n) = an , then
R∞
f (x)dx converges.
1
∞
X
an converges if and only if
n=1
3. (Ordinary Comparison Test) Suppose that 0 ≤ an ≤ bn for large enough n, then
(a) If
(b) If
∞
X
n=1
∞
X
bn converges, then
an diverges, then
n=1
∞
X
an converges.
n=1
∞
X
bn diverges.
n=1
4. (Limit Comparison Test) Suppose an ≥ 0 and bn > 0. Let L = lim
n→∞
(a) If 0 < L < ∞, then
∞
X
an and
n=1
(b) If L = 0 and
∞
X
∞
X
bn converge or diverge together.
n=1
bn converges, then
n=1
5. (Ratio Test) Let
then
∞
X
an
, then
bn
∞
X
an converges.
n=1
an+1
,
n→∞ an
an be a series where an > 0, compute the limit ρ = lim
n=1
(a) If ρ < 1, then the series converges.
(b) If ρ > 1, then the series diverges.
(c) If ρ = 1 the test is inconclusive.
6. (Alternating Series Test) If 0 < an+1 < an and lim an = 0, then the alternating series
n→∞
∞
X
(−1)n+1 an converges, otherwise the alternating series diverges.
n=1
7. (Absolute Convergence Theorem) If
8. (Absolute Ratio Test) Let
P
P
|un | converges, then
P
un converges.
|un+1|
,
n→∞ |un |
un be a series of nonzero terms and set ρ = lim
then we have the following:
(a) If ρ < 1, then the series converges absolutely (hence converges).
(b) If ρ > 1, then the series diverges
(c) If ρ = 1 then the test is inconclusive.
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