U of T · Analysis 1B · Winter

MAT159H1: integrals, series, convergence.

The second half of the analysis sequence: integration in depth, Taylor's theorem, infinite series, and the capstone idea of uniform convergence and power series.

About this course

Where analysis gets deep.

MAT159H1 is "Analysis 1B": integrals, the fundamental theorem, elementary transcendental functions, integration techniques, Taylor's theorem, series, uniform convergence, and power series. It continues from MAT158H1. This guide uses Spivak's Calculus as a rigorous companion. Uniform convergence is the key new idea and the bridge to later real analysis.

01

Chapters

In course order. Each page has definitions, theorems, worked proofs, pitfalls, and practice.

MAT159H1Analysis 1B · Winter MAT158H1Analysis 1A · prerequisite

  1. FTC, transcendentals & integration techniques

    FTC review, log/exp/trig foundations, substitution, by parts, partial fractions.

    Open →

  2. Improper integrals

    Unbounded intervals and integrands; convergence.

    Open →

  3. Sequences & Cauchy sequences

    Convergence and the Cauchy criterion.

    Open →

  4. Taylor's theorem

    Polynomial approximation and remainder bounds.

    Open →

  5. Infinite series

    Convergence tests; absolute vs conditional.

    Open →

  6. Uniform convergence

    Swapping limits with sums and integrals.

    Open →

  7. Power series

    Radius of convergence; term-by-term calculus.

    Open →
02

Textbook Spine

MAT159H1 continues the Spivak-style analysis habit: every computation is attached to a convergence or approximation claim. The public calendar confirms the topics; this guide uses Spivak's Calculus as the companion text and treats the current syllabus as the authority for exact assigned readings.

Unit-to-textbook alignment
Integration depth

Units 1-2 pair with the text's integration methods and improper-integral material. The goal is not only to evaluate an integral but to justify when the integral exists and which comparison proves convergence.

Sequences and approximation

Units 3-4 pair with sequence convergence and Taylor's theorem. Treat Cauchy arguments, monotonicity, boundedness, and remainder estimates as the core proof techniques.

Series and functions

Units 5-7 pair with infinite series, uniform convergence, and power series. The main theme is permission: when may limits, sums, derivatives, and integrals be interchanged?

READ

Read for conditions

Every theorem has hypotheses. Underline interval type, boundedness, monotonicity, positivity, uniformity, and differentiability assumptions.

TEST

Build counterexamples

For every convergence theorem, write one example showing why a missing hypothesis matters.

ESTIMATE

Practice bounds

Most MAT159 proofs are estimates. Keep a notebook of comparison, remainder, Cauchy, and uniform-convergence inequalities.

CONNECT

Link back to MAT158

FTC, continuity, compactness, and MVT never disappear; they become the justification layer under series and approximation.

03

Practice Plan

MAT159 practice should force decisions, not just formula recall. Each unit page gives local practice; use this structure for deeper weekly sets.

01

Technique first

For integration and series tests, write why the chosen method applies before doing any algebra.

02

Convergence second

For every improper integral or series, state whether it converges absolutely, conditionally, uniformly, or not at all.

03

Remainder control

Do Taylor problems with explicit error bounds, not only polynomial answers. The remainder is where proof meets approximation.

04

Mixed final sets

Combine one improper integral, one sequence, one Taylor estimate, one series test, one uniform-convergence proof, and one power-series problem.

Course-wide challenge bank

Use these as mixed checks after the local unit practice. Every answer should include the test or theorem that justifies the conclusion.

Challenge 1 · Improper integral

For which $p$ does $\displaystyle\int_e^\infty \dfrac{dx}{x(\ln x)^p}$ converge?

Solution sketch
Use $u=\ln x$, $du=dx/x$. The integral becomes $\int_1^\infty u^{-p}\,du$, which converges exactly when $p>1$.
Challenge 2 · Taylor error

Use Taylor's theorem to show that $|\sin x-(x-x^3/6)|\le |x|^5/120$ for all real $x$.

Solution sketch
Taylor's theorem with remainder through degree $3$ gives a fifth-order bound because the fourth-degree coefficient is zero and the fifth derivative is bounded in absolute value by $1$. Hence the remainder is at most $|x|^5/5!$.
Challenge 3 · Uniform convergence

Show $x^n\to0$ uniformly on $[0,a]$ when $0<a<1$, but not uniformly on $[0,1]$.

Solution sketch
On $[0,a]$, $\sup |x^n|=a^n\to0$. On $[0,1]$, the pointwise limit is $0$ for $x<1$ and $1$ at $x=1$, which is discontinuous; a uniform limit of continuous functions is continuous.
Challenge 4 · Power series

Find the radius of convergence of $\displaystyle\sum_{n=1}^{\infty}\dfrac{n^2x^n}{3^n}$.

Solution sketch
The ratio of consecutive absolute terms tends to $|x|/3$, so the series converges for $|x|<3$ and diverges for $|x|>3$. The radius is $3$.
04

Resources

MAT159H1 — official calendar listing
MAT158H1 guide — prerequisite
MAT137 guide — foundations

Companion text used by this guide: Spivak, Calculus. The public calendar confirms the MAT159H1 topic list but does not publish a current instructor textbook. Verify the assigned text, edition, and problem numbers against the active syllabus each year.