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{1}^{\infty}x e^{-x}dx=\Bigl[-x e^{-x}-e^{-x}\Bigr]{1}^{\infty}=\frac{2}{e}

  • Finite ⇒ series converges (by Integral Test).

“Continuous–Function Commutes with Limit” Trick

• Fact: If f is continuous and \lim{n\to\infty} an=L, then \lim{n\to\infty} f(an)=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{n\to\infty}an\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{n\to\infty}\frac{\frac{1}{1+n^2}}{\frac{1}{n^2}}=\lim{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{n=1}^{\infty}(-1)^{n-1} bn with b_n\ge0.

• Converges if

  1. b{n+1}\le bn (monotone decreasing)

  2. \displaystyle\lim{n\to\infty}bn=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 an converges but \sum|an| diverges.

• Hierarchy: Absolute ⇒ Conditional, proved via triangle inequality
  \Bigl|\sum{n=1}^N an\Bigr|\le \sum{n=1}^N |an|.

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: Rn=\left|\dfrac{a{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{n\to\infty}\left|\frac{(n+1)^3/3^{n+1}}{n^3/3^{n}}\right|=\frac13\lim{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{i=1}^N ci\Bigr|\le\sum{i=1}^N |ci|

• 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!