Lectures 14 and 15:
Infinite series, convergence tests, and power series
Definitions: sequence of partial sums, infinite sum. Infinite sums which converge,
diverge, or diverge to
±∞
.
Examples: geometric series,
∑
∞
n
=1
1
n
(
n
+1)
,
∑
√
n
+ 1

√
n
.
Definition: Cauchy criterion for infinite sums.
Proposition: If
∑
a
n
converges then
a
n
→
0.
Counterexample to show the converse of the previous proposition does not hold
(harmonic series).
Comparison test: Let
∑
a
n
be a series with
a
n
≥
0
∀
n
. If
∑
a
n
converges and

b
n
 ≤
a
n
for all
n
, then
∑
b
n
converges. If
∑
a
n
diverges to
∞
, and
b
n
≤
a
n
for all
n
, then
∑
b
n
diverges to
∞
.
Example:
∑
1
2
n
+1
.
Definition: absolute convergence.
Proposition: Absolutely convergent series are convergent.
Counterexample to show the converse of the previous proposition does not hold
(alternating harmonic series).
Ratio test: Suppose each
a
n
>
0.
1. If lim sup
a
n
+1
a
n
<
1, then the series converges absolutely.
2. If lim inf
a
n
+1
a
n
>
1, then the series diverges.
3. Otherwise, the test gives no information (i.e. there are series with both behav
iors).
Root test: Given the series
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