Lecture 3 Notes – Convergence Tests Deep-Dive

Course Logistics & Resources

• Practice materials now on course website

  • 2 practice exams uploaded

  • 1 is an actual past mid-semester exam (same length, topics & difficulty as upcoming test)

  • 1 additional mock exam

  • Extra weekly problem sets

  • Scans of textbook problems + some solutions

  • Full solutions will NOT be posted for every problem, but help is available:

    • Lecturer office hours

    • Maths Drop-in Centre

    • Demonstrator consultation sessions

• Timeline reminders

  • We are at the start of Week 3

  • Mid-semester exam: Monday of Week 6

  • Assignment 1 (≈ 4–5 multi-part questions)

  • Released end of this week / start of next

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

Toolbox of Series Convergence Tests (big picture)

• Previously covered: p-series, comparison, integral test

• Today’s focus (mostly examples):

  • Finishing Integral-Test examples

  • Divergence Test (a.k.a. “nth-term test”)

  • Direct & Limit Comparison

  • Continuous–function limit trick

  • Alternating-Series Test

  • Conditional vs Absolute convergence

  • Ratio Test

  • Root Test

Integral Test Example — $\sum_{k=1}^{\infty} k e^{-k}$

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

• Evaluate improper integral $\int_{1}^{\infty} x e^{-x}\,dx$

  • Integration by parts

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

  • After computation:
    $\int<em>{1}^{\infty}x e^{-x}dx=\Bigl[-x e^{-x}-e^{-x}\Bigr]</em>{1}^{\infty}=\frac{2}{e}$

  • Finite ⇒ series converges (by Integral Test).

“Continuous–Function Commutes with Limit” Trick

• Fact: If $f$ is continuous and $\lim<em>{n\to\infty} a</em>n=L$, then $\lim<em>{n\to\infty} f(a</em>n)=f(L)$.

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

  • Treat cube-root as continuous $f(x)=x^{1/3}$

  • First find inner limit:
    $\lim_{n\to\infty}\frac{n^2}{n^2+20n+9}=1$

  • Therefore overall term $\to f(1)=1$ (≠ 0) ⇒ fails Divergence Test ⇒ series diverges.

  • Key algebra: divide top & bottom by $n^2$ to reveal dominant terms.

Divergence (“nth-Term”) Test Refresher

• If $\displaystyle\lim<em>{n\to\infty}a</em>n\neq0$, then $\sum a_n$ diverges — no further work needed.

• Often fastest route to show divergence (called the “cheap” method in class).

Comparison & Limit-Comparison Examples

• Target series: $\sum_{n=1}^{\infty}\dfrac{1}{1+n^2}$

  • Looks like $\dfrac{1}{n^2}$ (a p-series with p=2>1 → convergent)

• Three possible strategies:

  1. Direct comparison: $\dfrac{1}{1+n^2}\le\dfrac{1}{n^2}$ ⇒ convergent.

  2. Limit Comparison (quickest):
     $\lim<em>{n\to\infty}\frac{\frac{1}{1+n^2}}{\frac{1}{n^2}}=\lim</em>{n\to\infty}\frac{n^2}{1+n^2}=1$ (finite ≠0) ⇒ same behaviour as $\sum\frac{1}{n^2}$ ⇒ convergent.

  3. Integral Test also works but is longer.

Alternating-Series Test (Leibniz Criterion)

• Form: $\sum<em>{n=1}^{\infty}(-1)^{n-1} b</em>n$ with $b_n\ge0$.

• Converges if

  1. $b<em>{n+1}\le b</em>n$ (monotone decreasing)

  2. $\displaystyle\lim<em>{n\to\infty}b</em>n=0$

• Classic example: Alternating harmonic series
  $\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n}=\ln 2$ (convergent even though harmonic series alone diverges).

Quick diagnostics on sample alternating series

Absolute vs Conditional Convergence

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

• Conditional: $\sum a<em>n$ converges but $\sum|a</em>n|$ diverges.

• Hierarchy: Absolute ⇒ Conditional, proved via triangle inequality
  $\Bigl|\sum<em>{n=1}^N a</em>n\Bigr|\le \sum<em>{n=1}^N |a</em>n|$.

Examples

• $a_n=(-1)^{n-1}\dfrac{1}{n}$: conditional (series converges, absolute diverges).

• $a_n=(-1)^{n-1}\dfrac{1}{n^2}$: absolutely (and hence conditionally) convergent.

• $a_n=(-1)^n\dfrac{3n}{4n-1}$: neither (fails absolute; fails alternating-series test).

Ratio Test

• Gadget: $R<em>n=\left|\dfrac{a</em>{n+1}}{a_n}\right|\xrightarrow[n\to\infty]{}L$

  • If L<1 ⇒ absolutely convergent

  • If L>1 ⇒ divergent

  • If $L=1$ ⇒ inconclusive

• Think of $L$ as the “effective ratio” vs a geometric series $\sum ar^n$.

Example 1

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

• Compute
  $L=\lim<em>{n\to\infty}\left|\frac{(n+1)^3/3^{n+1}}{n^3/3^{n}}\right|=\frac13\lim</em>{n\to\infty}\left(\frac{n+1}{n}\right)^3=\frac13$

• L=\tfrac13<1 ⇒ series absolutely converges.

Example 2 (factorials)

$\sum_{n=1}^{\infty}\frac{n^n}{n!}$

• Ratio
  L=\lim{n\to\infty}\frac{(n+1)^{n+1}/(n+1)!}{n^n/n!}=\lim{n\to\infty}\frac{(n+1)^{n}}{n^n}=\lim_{n\to\infty}\left(1+\frac1n\right)^n=e>1

• Divergent.

• Factorials & exponentials cancel neatly in ratios — ideal for this test.

Root Test

• Gadget: $\left|a_n\right|^{1/n}\xrightarrow[n\to\infty]{}L$

  • L<1 ⇒ absolute convergence

  • L>1 ⇒ divergence

  • $L=1$ ⇒ inconclusive

Example

$\sum_{k=2}^{\infty}\left(\frac{2k-1}{k^2+3}\right)^{!k}$

• Apply root test:
  L=\lim{k\to\infty}\left|\frac{2k-1}{k^2+3}\right|=\lim{k\to\infty}\frac{2k}{k^2}=\lim_{k\to\infty}\frac{2}{k}=0<1

• ⇒ Absolutely convergent (lecture example actually produced $L=2$ ⇒ divergence; both demonstrate method — watch dominant powers).

• Rule of thumb: any “something to the $k$” strongly suggests root test.

Key Inequalities & Identities Used

• Triangle inequality (finite sums): $\Bigl|\sum<em>{i=1}^N c</em>i\Bigr|\le\sum<em>{i=1}^N |c</em>i|$

• Factorial definition: $n! = n(n-1)(n-2)\dots1$ with $0! =1$

• Geometric-series sum (for |r|<1): $\sum_{n=0}^{\infty}ar^n=\dfrac{a}{1-r}$ (ratio tests mirror this behaviour).

Practical Strategy Checklist

• Always start by checking $\lim a_n$; if ≠0 ⇒ done (diverges).

• Look for obvious comparison to p-series or geometric series.

• If term includes $(–1)^n$ ⇒ try Alternating-Series Test.

• Factorials/exponentials ⇒ Ratio Test.

• Powers like $(\text{expression})^{n}$ ⇒ Root Test.

• When stuck, rewrite term: factor out largest $n^k$, simplify, or apply the continuous-function-commutes-with-limit trick.

• Remember absolute vs conditional: test $\sum|a_n|$ separately when signs alternate.

Recap: What Each Test Needs

• Integral Test: $f(x)\ge0$, decreasing, continuous; integrate $f(x)$.

• Comparison / Limit-Comparison: need a known benchmark series.

• Divergence Test: just the limit of $a_n$.

• Alternating-Series: monotone ↓ and limit zero on positive part $b_n$.

• Ratio / Root: compute limit of ratio or nth-root; compare with 1.

Prepare to mix & match — mastery comes from practising many examples (see added textbook scans & past exam). Happy studying!