Unit 13 · Term 2

Series

An infinite series is the limit of its partial sums. The art is choosing the right test to decide convergence — and remembering that a series can converge without converging absolutely.

By the end you can

Define convergence via partial sums; sum a geometric series.
Apply the divergence, integral, comparison, ratio, and root tests.
Use the alternating series test.
Distinguish absolute from conditional convergence.

Why it matters & where it comes from

Why it exists

How do you add infinitely many numbers and get something finite — and when can you? Series answer that, and they are how functions get turned into computable sums.

In the real world

Numerical libraries approximate functions such as $\sin$ and $e^x$ using polynomial and rational approximations closely related to series. Fourier methods underpin audio and image compression. Present-value calculations in finance are infinite sums, and error-correcting codes lean on related ideas.

Story & open questions

Oresme showed the harmonic series diverges around 1350 — but incredibly slowly. Euler stunned everyone in 1734 by proving $1+\tfrac14+\tfrac19+\cdots= \tfrac{\pi^2}{6}$ — why is $\pi$ here? And the deepest unsolved problem in mathematics, the Riemann Hypothesis (a one-million-dollar Millennium Prize), is a statement about a series that secretly controls the prime numbers.

Choosing a series test

Series questions are decision problems. Before calculating, identify the structure of the term $a_n$. A good test choice should match that structure.

Test selection guide

If $a_n$ does not tend to $0$, use the divergence test.

If the series is geometric or telescoping, compute partial sums directly.

If terms look like powers of $n$, compare with a $p$-series.

If factorials or exponentials appear, try the ratio test.

If the entire term is raised to the $n$th power, try the root test.

Absolute versus conditional convergence

For signed series, first ask whether the absolute series converges. Absolute convergence is stronger and usually easier to work with. If the absolute series diverges, then look for alternating structure or another cancellation argument.

Absolute convergence: $\sum|a_n|$ converges, so $\sum a_n$ converges.
Conditional convergence: $\sum a_n$ converges but $\sum|a_n|$ diverges.
Divergence: either the terms do not go to $0$, or every applicable test shows the partial sums cannot settle.

Partial sums and geometric series

Definition

$\displaystyle\sum_{n=1}^\infty a_n$ converges to $S$ if the partial sums $s_N=\sum_{n=1}^N a_n$ converge to $S$.

Geometric series

$\displaystyle\sum_{n=0}^\infty r^n=\frac{1}{1-r}$ for $|r|<1$; diverges for $|r|\ge1$.

The divergence test

Theorem

If $\sum a_n$ converges then $a_n\to0$. Contrapositive: if $a_n\not\to0$, the series diverges. (The converse is false — see the harmonic series.)

Tests for positive series

Toolkit

Integral test: $\sum f(n)$ and $\int_1^\infty f$ converge or diverge together when $f$ is continuous, positive, and decreasing. Comparison / limit comparison: compare with a known series. Ratio: $L=\lim|a_{n+1}/a_n|$. Root: $L=\lim|a_n|^{1/n}$. For either test, $L<1$ gives absolute convergence, $L>1$ (or $L=\infty$) gives divergence, and $L=1$ is inconclusive.

Example $p$-series

By the integral test, $\displaystyle\sum\frac1{n^p}$ converges iff $p>1$. So $\sum1/n^2$ converges, $\sum1/n$ (harmonic) diverges.

Alternating series and absolute convergence

Alternating series test

If $b_n\downarrow0$, then $\sum(-1)^n b_n$ converges.

Definition

$\sum a_n$ converges absolutely if $\sum|a_n|$ converges (which implies convergence). If it converges but not absolutely, it is conditional.

Example

$\displaystyle\sum\frac{(-1)^n}{n}$ converges (alternating) but not absolutely ($\sum1/n$ diverges) — conditional.

Common pitfalls

• $a_n\to0$ does not imply convergence (harmonic series).
• The ratio test is inconclusive when $L=1$ — switch tests.
• Check the hypotheses of the alternating test: terms decreasing and tending to $0$.

Practice

Problems adapted from the MAT137 question bank by Alfonso Gracia-Saz, used under CC BY-SA 4.0. Write full solutions before opening the answer.

Practice 1 · Comparison

Does $\displaystyle\sum_{n=1}^\infty\frac{1}{n^2+1}$ converge?

Answer

Yes, by comparison with $\sum1/n^2$.

Practice 2 · Ratio test

Test $\displaystyle\sum_{n=1}^\infty\frac{n}{2^n}$.

Answer

Ratio: $\dfrac{a_{n+1}}{a_n}=\dfrac{n+1}{2n}\to\tfrac12<1$, converges.

Practice 3 · Conditional

Is $\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt n}$ absolutely or conditionally convergent?

Answer

Conditional: alternating test gives convergence, but $\sum1/\sqrt n$ ($p=\tfrac12$) diverges.

Practice 4 · Geometric

Sum $\displaystyle\sum_{n=0}^\infty\frac{3}{4^n}$.

Answer

Geometric, $r=\tfrac14$: $\dfrac{3}{1-1/4}=4.$

Practice 5 · Telescoping

Find $\displaystyle\sum_{n=1}^\infty\frac{1}{n^2+2n}$ via partial sums.

Solution

$\dfrac1{n(n+2)} =\tfrac12\!\left(\dfrac1n-\dfrac1{n+2}\right)$. The partial sums telescope to $\tfrac12\!\left(1+\tfrac12\right)=\tfrac34.$

Practice 6 · p-series

For which $p$ does $\displaystyle\sum_{n=1}^\infty\frac1{n^p}$ converge? What about $\sum 1/n$?

Answer

Converges iff $p>1$. The harmonic series $\sum1/n$ ($p=1$) diverges.

Practice 7 · Comparison

Determine convergence of $\displaystyle\sum_{n=1}^\infty\frac{n^2+3n}{n^4+5n+1}$.

Answer

Terms behave like $\dfrac{n^2}{n^4}=\dfrac1{n^2}$; by limit comparison with $\sum1/n^2$ it converges.

Practice 8 · Ratio test

Test $\displaystyle\sum_{n=1}^\infty\frac{3^n}{n!}$.

Answer

Ratio $\dfrac{3^{n+1}/(n+1)!}{3^n/n!}=\dfrac{3}{n+1}\to0<1$, so it converges (to $e^3-1$).

Expanded topic checklist

Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:

Series definition. Understand a series as the limit of partial sums.
Basic tests. Use divergence, geometric, comparison, limit comparison, integral, ratio, and root tests.
Alternating series. Use alternating-series estimates and distinguish conditional from absolute convergence.
Strategy. Choose tests by recognizing the algebraic, exponential, factorial, or alternating structure of the terms.

Textbook/source alignment

Use the Alfonso Gracia-Saz MAT137 materials as the primary local spine, with the current UofT single-variable calculus topic split as the completeness check. The added items are original study prompts, not copied textbook exercises.

Core theorem map

The series spine is: analyze the term, choose a test, prove the hypotheses, and state whether convergence is absolute, conditional, or divergent.

Study routine. When a series has signs, test absolute convergence first; only then use alternating or cancellation-based reasoning.

Deep practice extensions

Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.

Extension 1 · Divergence test

What does the term test say about $\sum n/(n+1)$?

Guide

The terms tend to $1$, not $0$, so the series diverges.

Extension 2 · Comparison

Determine whether $\sum_{n=1}^\infty 1/n^2$ converges.

Guide

It is a p-series with $p=2>1$, so it converges.

Extension 3 · Alternating

Show $\sum (-1)^{n+1}/n$ converges but not absolutely.

Guide

The alternating terms decrease to $0$, so the series converges by the alternating test. The absolute series is harmonic and diverges.

Extension 4 · Ratio test

Apply the ratio test to $\sum n!/3^n$.

Guide

The ratio is $(n+1)!/3^{n+1}\cdot 3^n/n!=(n+1)/3\to\infty$, so the series diverges.

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