Infinite Series
Adding infinitely many numbers — when it makes sense and how to test it. The convergence tests here are the working vocabulary for the uniform convergence and power series that close the course.
How can an infinite sum be finite, and when? Series make this precise by reducing the question to a sequence — the partial sums — and let functions be represented as infinite sums, which is where the course is headed.
Calculators sum series to find $\sin$, $e^x$, and $\pi$ to any precision; Fourier series compress audio and images (MP3, JPEG); present value in finance is an infinite geometric sum.
Euler stunned everyone in 1734 with $1+\tfrac14+\tfrac19+\cdots= \tfrac{\pi^2}{6}$ (the Basel problem). The deepest open problem in mathematics — the Riemann Hypothesis, a one-million-dollar Millennium Prize problem — is a statement about where a series related to the primes, $\sum n^{-s}$, vanishes.
- Define convergence by partial sums; sum a geometric series.
- Apply the divergence, comparison, limit comparison, integral, ratio, and root tests, and know when each one applies.
- Use the alternating series test and tell absolute from conditional convergence.
- Explain why rearranging a conditionally convergent series can change its sum.
Convergence and the basic tests
$\sum a_n$ converges if the sequence of partial sums $s_N=\sum_{n=1}^{N}a_n$ converges. Geometric series: $\sum_{n\ge0}r^n=\tfrac{1}{1-r}$ for $|r|<1$, and diverges for $|r|\ge1$.
If $a_n\not\to0$, then $\sum a_n$ diverges. (Contrapositive of: convergent series have terms tending to $0$.) The converse is false — this is the single most common mistake in the subject.
For $a_n,b_n\ge0$: if $a_n\le b_n$ and $\sum b_n$ converges, so does $\sum a_n$. If instead $\lim a_n/b_n=L$ with $0<L<\infty$, then $\sum a_n$ and $\sum b_n$ converge or diverge together — the series analogue of the comparison tests from improper integrals in Unit 2.
The integral, ratio, and root tests
If $f$ is positive, continuous, and decreasing on $[1,\infty)$ with $f(n)=a_n$, then $\sum a_n$ converges iff $\int_1^\infty f(x)\,dx$ converges — series and improper integrals rise or fall together, since the partial sums are squeezed between the integral and itself shifted by one term.
$\sum n^{-p}$ converges iff $p>1$, by the integral test applied to $\int x^{-p}dx$ — exactly the $p$-test from Unit 2, now for sums.
Let $L=\lim\left|\dfrac{a_{n+1}}{a_n}\right|$ or $L=\lim\sqrt[n]{|a_n|}$. If $L<1$ the series converges absolutely; if $L>1$ it diverges; if $L=1$ neither test decides anything.
$\sum\dfrac{n}{2^n}$: $\dfrac{a_{n+1}}{a_n}=\dfrac{n+1}{2n}\to\tfrac12<1$, so it converges.
Alternating and absolute convergence
If $b_n\downarrow0$ (decreasing to $0$), then $\sum(-1)^n b_n$ converges, and the error from truncating after $N$ terms is at most $b_{N+1}$.
If $\sum|a_n|$ converges, $\sum a_n$ converges absolutely (and hence converges, by comparison with $|a_n|$). A series that converges but not absolutely converges only conditionally.
$\sum\tfrac{(-1)^n}{n}$ converges (alternating test) but $\sum\tfrac1n$ diverges ($p=1$) — so it converges conditionally, not absolutely.
• $a_n\to0$ does not imply convergence — the harmonic series
$\sum1/n$ is the standard counterexample.
• The ratio and root tests are inconclusive at $L=1$; switch to another
test.
• The integral test needs $f$ eventually positive, continuous, and
decreasing — check all three.
• Check both hypotheses of the alternating series test (decreasing and
$\to0$) — a non-monotone alternating sequence can fail even if its terms
$\to0$.
• Never reorder the terms of a series you only know converges conditionally.
Practice
Does $\sum\dfrac{1}{n^2+1}$ converge?
Answer
Test $\sum\dfrac{n}{2^n}$ (ratio test).
Answer
Absolute or conditional: $\sum\dfrac{(-1)^n}{\sqrt n}$?
Answer
Test $\displaystyle\sum\left(\dfrac{n}{n+1}\right)^{n^2}$.
Solution
Does $\sum_{n\ge2}\dfrac{1}{n\ln n}$ converge?
Solution
Estimate $\sum_{n=1}^\infty\dfrac{(-1)^{n+1}}{n^2}$ by its first 3 terms, and bound the error.
Answer
Test $\sum\dfrac{3n^2+1}{n^4+2n}$.
Solution
The alternating harmonic series $\sum(-1)^{n+1}/n=\ln2$. Explain (without computing it) why summing "two positive terms, then one negative" in a different order can give a different total.
Answer
Expanded topic checklist
Course coverage. This unit is part of the complete MAT159H1 topic map. It should cover the following items before you move on:
- Partial sums. Define series convergence through convergence of partial sums.
- Positive series. Use comparison, limit comparison, integral, ratio, and root tests.
- Signed series. Use absolute convergence and alternating series tests.
- Error estimates. Estimate tails for geometric, alternating, and integral-test series.
Use the current UofT MAT159H1 calendar topics as the coverage spine, and pair lecture notes with a Spivak-style proof-calculus text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
The series spine is: first test whether terms go to zero, then select a convergence test based on term structure, then estimate the tail if needed.
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.
For what $x$ does $\sum x^n$ converge?
Guide
Use the integral test on $\sum 1/(n\ln n)$ for $n\ge2$.
Guide
Apply the root test to $\sum (2n/(3n+1))^n$.
Guide
How many terms guarantee the alternating harmonic sum is within $0.01$ of its limit?