Topic 5 - Series

Question 1

Infinite series

Answer 1

The sum of all terms in a sequence

Question 2

Partial series

Answer 2

The sum of all terms in a sequence up to N ∈ IN.

Question 3

Absolute convergence

Answer 3

A series converges absolutely if:

∑ |a_n| < ∞

Question 4

Conditional convergence

Answer 4

A series converges conditionally if:
∞
∑ a_n converges and
n

∞
∑ |a_n| diverges.
n

Question 5

Comparison Test for Series

Answer 5

Suppose that (a_n) and (b_n) are real sequences and|a_n|≤ b_n for all n∈N. If:
∞
∑ b_n < ∞, then
n=1
∞
∑ a_n converges absolutely.
n=1

Question 6

D’Alambert Ratio Test

Answer 6

Suppose that (a_n) is a real sequence and a_n ≠ 0 for all n, and

r = lim (|a_{n+1}|/|an|)
n→∞

exists (where r=∞ is allowed). If:
i. r<1, then the series (a_n) converges absolutely.
ii. r>1, then the series (a_n)diverges

Question 7

Cauchy Condensation Test

Answer 7

Suppose (a_n) is a decreasing sequence with (a_n) ≥ 0 for all n. Let (b_k) = 2^{k} a_(2^{k}) for all k. Then:
∞ ∞
∑ |a_n| < ∞ if and only if ∑ |b_k| < ∞ .
n=1 k=1

Question 8

Leibniz Alternating Series Test

Answer 8

Suppose (a_n) is a decreasing series such that (a_n) → 0 as n → ∞. Then the series:
∞
∑ (-1)^{n} a_n converges
n

Question 9

Radius of Convergence

Answer 9

Given a power series, then:

R = lim (|a_n|/|a_{n+1}|)
n→∞

Question 10

Cauchy-Hadamard Theorem

Answer 10

R = 1 / limsup (|an|)^{1/n}
n→∞