Series Convergence Tests — Lecture 3 Detailed Notes

Course Logistics

• Practice material now online

  • 1 authentic past mid-semester exam + 1 extra practice paper

  • Format, breadth & difficulty mirror upcoming Week-6 mid-semester exam

• Weekly extra problems

  • Scans of textbook questions + selected solutions

  • Full solutions not posted → ask in office hours, drop-in centre, or demonstrator consultation

• Assignment 1

  • Released end of Week 3 / start of Week 4

  • 4–5 questions, each with multiple parts

  • Due sometime in Week 7 (after the mid-semester exam)

Lecture Focus

• Few new theorems; mostly practice & technique

• Complete leftover examples and add:

  • Alternating Series Test

  • Ratio Test

  • Root Test

  • Absolute vs Conditional Convergence

Integral Test (catch-up example)

• Series: $\sum_{k=1}^{\infty} k e^{-k}$

• Analogous function: $f(x)=x e^{-x}$ (positive, continuous, decreasing for $x\ge1$)

• Evaluate improper integral: $\int_1^{\infty} x e^{-x}\,dx$

  • Integration by parts

  • Let $u=x\,,\; dv=e^{-x}dx\Rightarrow v=-e^{-x}$

  • Result after limits: $\frac{2}{e}$ (finite)

• Integral converges ⇒ series converges by Integral Test

Key Technique Reminder

• For rational functions inside limits: factor highest power of $n$ in numerator & denominator  leftover fractions $\to0$.

“Continuous Function commutes with limit” Trick

• If $f$ is continuous and $\lim_{n\to\infty}a_n=L$, then $\lim_{n\to\infty}f(a_n)=f(L)$.

• Example series: $\sum_{n=1}^{\infty} \sqrt[3]{ \frac{n^2}{n^2+20n+9} }$

  • Use cubic-root continuity: pull limit inside root

  • $\lim_{n\to\infty} \frac{n^2}{n^2+20n+9}=1$ ⇒ outer limit $=1$

  • Since term limit $\neq0$ ⇒ series diverges by Divergence Test (a.k.a. $n\to\infty$ term test).

Choosing a Comparison (Example $\sum \frac1{1+n^2}$)

• Eyeball dominant term $\sim\frac1{n^2}$ (p-series with p=2>1 ⇒ expect convergence).

1. Direct Comparison: $\frac1{1+n^2} \le \frac1{n^2}$.

2. Limit Comparison (slicker):
   $\lim_{n\to\infty}\frac{ \frac1{1+n^2} }{ \frac1{n^2} }=1$ ⇒ same behaviour ⇒ convergent.

3. Integral Test also works but more labour.

Alternating Series Test (Leibniz Criterion)

• Series of form $\sum_{n=1}^{\infty} (-1)^{n-1} b_n$ with

  1. b_n>0

  2. $b_{n+1}\le b_n$ (monotonically decreasing)

  3. $\lim_{n\to\infty} b_n =0$
     ⇒ Series converges (actually conditionally unless also absolutely conv.).

Classic Example: Alternating Harmonic Series

$\sum_{n=1}^{\infty} (-1)^{n-1}\frac1n = \ln 2$ (convergent although $\sum 1/n$ diverges).

Counter-Examples / Quick Checks

• If either monotonicity or $b_n\to0$ fails → cannot apply test; often diverges by $\lim a_n$ test.

  • Example $a_n=(-1)^n\frac{3n}{4n-1}$: $b_n\not\to0$ ⇒ diverges.

  • Example $(-1)^n \frac{n^2}{n^2+1}$: monotone ↑ and $\lim b_n=1$ ⇒ diverges.

Absolute vs Conditional Convergence

• Absolute Convergence: $\sum |a_n|$ converges.

• Conditional Convergence: Original series converges but $\sum |a_n|$ diverges.

• Relation: $\text{Absolute} \Rightarrow \text{Convergent}$ by Triangle Inequality
  $\Big|\sum_{n=1}^{N} a_n\Big|\le\sum_{n=1}^{N}|a_n|$.

Examples

1. $(-1)^{n}\dfrac{3n}{4n-1}$ – not abs. (limit ≠0) and not cond. ⇒ diverges.

2. Alternating harmonic $(-1)^{n-1}/n$ – convergent but not absolute ⇒ conditionally convergent.

3. $(-1)^{n-1}/n^2$ – $p=2$ so abs. conv. ⇒ both absolute & conditional.

Ratio Test

• Define $R =\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$.

  • If R<1 ⇒ absolutely convergent.

  • If R>1 ⇒ divergent.

  • If $R=1$ ⇒ inconclusive.

• Heuristic: compares with geometric series (common ratio $R$).

Good when series contains factorials, exponentials, large powers.

Example 1

$a_n = (-1)^n\frac{n^3}{3^n}$

• $\left|\frac{a_{n+1}}{a_n}\right|=\frac{(n+1)^3}{3^{n+1}}\cdot\frac{3^{n}}{n^3}=\frac{(n+1)^3}{3 n^3}$

• Pull $n^3$: \to\frac13\Big(1+\frac1n\Big)^3 \xrightarrow[n\to\infty]{}\frac13<1 ⇒ absolutely convergent.

Example 2 (factorial)

$a_n=\frac{n^n}{n!}$

• $\left|\frac{a_{n+1}}{a_n}\right|=\frac{(n+1)^{n+1}}{(n+1)!}\cdot\frac{n!}{n^n}=\frac{(n+1)^{n}}{n^n}\cdot\frac{1}{n+1}$

• Simplify: \Big(1+\frac1n\Big)^n\cdot\frac1{n+1}\to e\cdot0 = e>1

• Actually final ratio =e>1 ⇒ diverges.

Root Test

• Define $L=\lim_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}|a_n|^{1/n}$.

  • L<1 ⇒ absolutely convergent.

  • L>1 ⇒ divergent.

  • $L=1$ ⇒ inconclusive.

• Often simpler when term already has nth power.

Example

$a_k=\left( \dfrac{2k-1}{k^2+3} \right)^{!k}$, $k\ge2$

• Root test: $L=\lim_{k\to\infty}\left|a_k\right|^{1/k}=\lim_{k\to\infty}\frac{2k-1}{k^2+3}$

• Simplify: factor $k$/$k^2$ ⇒ L=\frac{2}{1}=2>1 ⇒ diverges.

How to Select a Test (Heuristics)

• Divergence test first: if $\lim a_n\neq0$ ⇒ stop (diverges).

• p-series / geometric resemblance ⇒ comparison or limit comparison.

• Positive, monotone, integrable function ⇒ integral test.

• Alternating signs ⇒ try Alternating Series Test.

• Factorials, exponentials, $n^n$ ⇒ Ratio Test.

• $a_n^n$ pattern ⇒ Root Test.

• When absolute values easier ⇒ decide absolute vs conditional.

Ethical / Practical Notes

• Copyright notice: material reproduced under Section 113P, further reproduction prohibited.

• Use provided past exams & extra problems responsibly; seek help but do not share full solutions online.

Quick Formula & Reference Sheet

• Triangle Inequality (finite sums): $\Big|\sum_{k=1}^N x_k\Big|\le\sum_{k=1}^N |x_k|$

• Geometric series: \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\,(|r|<1)

• p-series: $\sum_{n=1}^{\infty} \frac1{n^p}$ converges \iff p>1.

• Stirling (handy for factorial limits): $n!\sim\sqrt{2\pi n}\,(n/e)^n$ (not used directly today but useful with Ratio/Root).

Big Take-aways

• Master pattern recognition: decide test before heavy algebra.

• Always check the easy tests (divergence, comparison) first.

• Alternation aids convergence; absolute convergence is a stronger requirement.

• Ratio & Root tests deliver fast verdicts when cancellation of large growth terms occurs.

• Show all limit steps clearly; most proofs hinge on correct limit evaluation.