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**Unformatted text preview: **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 ±∞.
√
√
n + 1 − n.
Examples: geometric series, ∞ n(n1+1) ,
n=1
Deﬁnition: Cauchy criterion for inﬁnite sums.
Proposition: If
an converges then an → 0.
Counterexample to show the converse of the previous proposition does not hold
(harmonic series).
Comparison test: Let
an be a series with an ≥ 0 ∀n. If
an converges and
|bn | ≤ an for all n, then
bn converges. If
an diverges to ∞, and bn ≤ an for all
n, then
bn diverges to ∞.
1
.
Example:
2n +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 an > 0.
1. If lim sup
2. If lim inf an+1
an
an+1
an < 1, then the series converges absolutely.
> 1, then the series diverges. 3. Otherwise, the test gives no information (i.e. there are series with both behaviors).
an , let α = lim sup |an |1/n . Root test: Given the series 1. If α < 1, then the series converges absolutely.
2. If α > 1, then the series diverges.
3. Otherwise α = 1 and the test gives no information.
n
n!
1
1
,
,
,
.
Examples:
2n
2n
n2
n
Integral test: Let f be a continuous function deﬁned on [0, ∞) that is both positive
and decreasing. Then
f (n) converges iﬀ
n lim n→∞ exists as a real number.
Application: the p-series 1
np f (x)dx
1 converges iﬀ p > 1.
1 Alternating series test: If (an ) is decreasing and an > 0 for all n, and if lim an = 0,
then (−1)n an converges.
cos(nπ )
√
Examples: alternating harmonic series,
.
n
Deﬁnition: power series, radius of convergence.
Ratio criterion: The radius of convergence R of a power series
an xn is equal to
an
lim an+1 , provided the limit exists.
Examples: xn , 1n
x,
n 1n
x.
n! 2 ...

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