Sequences, Series, and Tests

How do you test if a series converges?

Compute lim_n→∞ a_n

if limit = finite number, series converges

if limit = ±∞ or DNE, series diverges

Squeeze Theorem for Sequences

if:

a_n ≤ c_n ≤ b_n

and:

lim a_n = lim b_n = L

then:

lim c_n = L

Definition of a series…

∑a_n = lim_n→∞ ∑ⁿ_{k = 1} a_k

you are checking for partial sums

Geometric Series Test

Form: ∑arⁿ or ∑ar^n-1

Condition: | r | < 1

Result: ∑arⁿ = a / (1 - r)

if: | r | ≥ 1, then divergent

Divergence Test (nth-term test)

check: lim_n→∞ a_n

if limit ≠ 0, then diverges

if limit = 0, inconclusive

Integral Test (When do you use it?)

You need ALL 3:

f(x) = a_n

positive, continuous, decreasing

Integral Test - What does it actually mean and how do I use it?

Step 1 (Pattern Recognition): ∑a_n = ∑ f(n)

Step 2 (Turn it into an Integral) ∫^∞₁ f(x) dx

Step 3:

If the integral gives a finite number = convergent

if the integral = ∞ (blows up) = divergent

Direct Comparison Test — When to use?

Use When:

Terms look similar to known series

Usally compare with, 1 / (n^p) or geometric.

Direct Comparison — Conditions

for a_n, b_n > 0…

if: a_n ≤ b_n

and ∑b_n converges → ∑a_n converges

if: a_n ≥ b_n

and ∑b_n diverges → ∑a_n diverges

Limit Comparison Test — When to Use?

Use when:

Fractions with similar powers

hard to compare directly

Limit Comparison — Result

Compute: lim_n→∞ (a_n/b_n) = L

if 0 < L < ∞ → SAME Behavior (Both converge or Both diverge)

Alternating Series Test (AST) — Conditions

Series: ∑(-1)ⁿb_n or (-1)ⁿ⁺¹b_n

You need: b_n decreasing and lim b_n = 0

AST - What does it Prove?

Series Converges (but only conditionally)

Absolute vs Conditional Convergence

if….

∑|a_n| converges, then absolutely convergent

if…

∑a_n converges but ∑|a_n| diverges, then conditionally convergent

Ratio Test — When to use?

Best for:

Factorials

Exponentials

Powers like nⁿ

Ratio Test — formula

L = lim_n→∞ |(a_n+1/a_n)|

Ratio Test — Result

L < 1 → Converges (Absolutely)

L > 1 → Diverges

L = 1 → Inconclusive

Root Test — When to use

Best for:

powers like (a_n)ⁿ

Root Test — Formula

L = lim_n→∞ ⁿ√|a_n|

Root Test — Result

L < 1 → Converges (Absolutely)

L > 1 → Diverges

L = 1 → Inconclusive

p-Series Test

∑(1/n^p)

p > 1 Converges

p ≤ 1 → Diverges

Harmonic Series

∑1/n → Diverges

Divergence Test Formula

lim _n→∞ a_n ≠ 0 → Diverges

Ratio Test Formula

L = lim_n→∞ |(a_n+1/a_n)|

Root Test Formula

L = lim_n→∞ ⁿ√|a_n|

Limit Comparison Formula

lim_n→∞ (a_n/b_n) = L

Integral Test Setup

∑a_n ↔ ∫^∞₁ f(x) dx

Alternating Series Form

∑(-1)ⁿ b_n or (-1)ⁿ⁺¹ b_n