Calculus: Sequences and Series flashcards

Question 1

What is a sequence?

Accepted Answer

An ordered list of numbers a₁, a₂, a₃, …

Question 2

What is the index of a sequence?

Accepted Answer

The subscript used to define its terms

Question 3

What is a recurrence relation?

Accepted Answer

A rule defining a term aₙ of a sequence by earlier terms

Question 4

What is an arithmetic sequence?

Accepted Answer

A sequence where every pair of consecutive terms has the same difference

Question 5

What is a geometric sequence?

Accepted Answer

A sequence where the ratio aₙ₊₁/aₙ is the same for all positive integers n

Question 6

What is a monotone sequence?

Accepted Answer

A sequence that is increasing or decreasing

Question 7

What does a bounded, monotone sequence always do?

Accepted Answer

It converges

Question 8

For which r does the geometric sequence {rⁿ} converge?

Accepted Answer

Exactly when |r| < 1 or r = 1

Question 9

What is the kth partial sum of the series Σaₙ?

Accepted Answer

S_k = a₁ + a₂ + ⋯ + a_k

Question 10

When does an infinite series converge?

Accepted Answer

When its sequence of partial sums converges

Question 11

Sum of the geometric series Σ arⁿ⁻¹ (n from 1) when |r| < 1?

Accepted Answer

a/(1 − r)

Question 12

When does the geometric series Σ arⁿ⁻¹ diverge?

Accepted Answer

When |r| ≥ 1

Question 13

Does the harmonic series 1 + 1/2 + 1/3 + ⋯ converge or diverge?

Accepted Answer

It diverges

Question 14

What is a telescoping series?

Accepted Answer

A series in which most terms cancel in each partial sum

Question 15

Divergence test: what follows if lim aₙ ≠ 0 as n → ∞?

Accepted Answer

The series Σaₙ diverges

Question 16

If lim aₙ = 0, what can be said about Σaₙ?

Accepted Answer

Nothing certain: it may converge or diverge

Question 17

Integral test: if aₙ > 0 and f is continuous and decreasing with f(n) = aₙ, how do Σaₙ and ∫₁^∞ f(x) dx relate?

Accepted Answer

They either both converge or both diverge

Question 18

For which p does the p-series Σ 1/nᵖ converge?

Accepted Answer

Converges if p > 1; diverges if p ≤ 1

Question 19

Comparison test: if 0 ≤ aₙ ≤ bₙ for all n ≥ N and Σbₙ converges, what about Σaₙ?

Accepted Answer

Σaₙ converges

Question 20

Alternating series test: when does an alternating series converge?

Accepted Answer

When bₙ₊₁ ≤ bₙ for all integers n ≥ 1 and bₙ → 0

Question 21

When does Σaₙ converge absolutely?

Accepted Answer

When Σ|aₙ| converges

Question 22

When does Σaₙ converge conditionally?

Accepted Answer

When Σaₙ converges but Σ|aₙ| diverges

Question 23

Ratio test for Σaₙ with nonzero terms: what do ρ = lim |aₙ₊₁/aₙ| values tell you?

Accepted Answer

0 ≤ ρ < 1: converges absolutely; ρ > 1: diverges; ρ = 1: inconclusive

Question 24

Root test: what is ρ, and what does ρ < 1 tell you?

Accepted Answer

ρ = lim of the nth root of |aₙ|; if 0 ≤ ρ < 1, the series converges absolutely

Calculus: Sequences and Series flashcards

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