Mathfolis

Comparison Tests

Unit 10 · Infinite Sequences & Series

p-Series and Comparison Tests

A p-series ∑1/nᵖ converges if p > 1 and diverges if p ≤ 1. The harmonic series (p = 1) diverges even though its terms go to 0 — memorize this. For other series, the Direct Comparison Test (bound term-by-term) and Limit Comparison Test (compare the dominant-term ratio) determine convergence by relating to a known series.

p-Series

∑n=1∞1np  {convergesp>1divergesp≤1\sum_{n=1}^{\infty}\frac{1}{n^p}\;\begin{cases}\text{converges} & p>1\\\text{diverges} & p\le 1\end{cases}

Limit Comparison Test (LCT)

If lim⁡n→∞anbn=c,  0<c<∞, then ∑an and ∑bn behave the same.\text{If } \lim_{n\to\infty}\frac{a_n}{b_n}=c,\;0<c<\infty, \text{ then } \sum a_n \text{ and } \sum b_n \text{ behave the same.}

Direct Comparison Test (DCT)

If 0 ≤ aₙ ≤ bₙ and ∑bₙ converges, then ∑aₙ converges.
If 0 ≤ aₙ ≤ bₙ and ∑aₙ diverges, then ∑bₙ diverges.
Caution: The harmonic series ∑1/n diverges even though 1/n → 0. This is the most important counterexample in series.
AP Tip: For the LCT, choose bₙ by keeping only the dominant power of n in the numerator and denominator.
Type 1

p-Series Classification

Rewrite the series in the form ∑1/nᵖ and apply the p-series rule.

Example 1
Determine whether each series converges or diverges: (a) ∑1/n(3/2)n^{(3/2)} (b) ∑1/∛n.
Type 2

Limit Comparison Test

Choose bₙ from the dominant terms, compute lim(aₙ/bₙ), and classify ∑bₙ.

Example 2
Determine whether ∑(3n² + 1)/(n⁴ − 2) converges or diverges.
Type 3

Direct Comparison Test

Bound the series term-by-term above (for convergence) or below (for divergence) by a known convergent or divergent series.

Example 3
Determine whether ∑1/(2ⁿ + n) converges or diverges.
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