The finale: functions written as infinite polynomials. Power series let us represent, differentiate, integrate, and approximate functions — and Taylor's theorem tells us how good the approximation is.
A power series rewrites a complicated function as an infinite polynomial — something you can actually compute, differentiate, integrate, and approximate with a known error.
Power-series ideas are one way computers approximate functions; they also explain how physics gets workable approximations (the small-angle $\sin\theta\approx\theta$ that makes pendulums solvable); and how optimization methods like Newton's work. Truncated Taylor series are everywhere in applied math and machine learning.
Around 1400, Madhava in Kerala, India, discovered the series for sine, cosine, and arctangent — including a formula for $\pi$ — three centuries before Newton and Leibniz. From these same ideas comes Euler's identity $e^{i\pi}+1=0$, routinely voted the most beautiful equation in mathematics.
A power series has two separate questions: where does it converge, and what function does it represent there? The radius handles the first question in the interior; endpoint checks and remainders handle the rest.
Do not compute every Taylor series from derivatives. Most MAT137 problems are faster if you start from a known series and transform it carefully.
A power series $\displaystyle\sum_{n=0}^\infty c_n(x-a)^n$ converges absolutely for $|x-a|<R$ and diverges for $|x-a|>R$. When $R$ is finite, the endpoints $|x-a|=R$ must be checked separately. The radius $R$ is often found by the ratio or root test.
$\displaystyle\sum_{n=1}^\infty\dfrac{x^n}{n}$: ratio $\left|\dfrac{x^{n+1}/(n+1)}{x^n/n}\right|\to |x|$, so $R=1$. Endpoints: converges at $x=-1$, diverges at $x=1$; interval $[-1,1)$.
The Taylor series of $f$ about $a$ is $\displaystyle\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n.$ About $a=0$ it is the Maclaurin series.
$e^x=\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}$ (all $x$), $\sin x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ (all $x$), $\dfrac{1}{1-x}=\displaystyle\sum_{n=0}^\infty x^n$ ($|x|<1$), $\ln(1+x)=\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}x^n}{n}$ ($-1<x\le1$).
If $f$ has $n+1$ continuous derivatives on an interval containing $a$ and $x$, then $f(x)=\displaystyle\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k+R_n(x)$ where the Lagrange remainder is $R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$.
Approximate $\cos(0.1)$ by its degree-3 Taylor polynomial $1-\tfrac{x^2}{2}$ (the cubic coefficient is zero). The Lagrange remainder bounds the error by $\dfrac{(0.1)^4}{4!}\approx4.2\times10^{-6}$.
Inside its radius of convergence a power series may be differentiated and integrated term by term, with the same radius.
Integrating $\dfrac{1}{1+x}=\displaystyle\sum_{n=0}^\infty(-x)^n$ gives $\ln(1+x)=\displaystyle\sum_{n=1}^\infty\dfrac{(-1)^{n+1}x^n}{n}$ for $|x|<1$.
• Always check the endpoints separately — the radius doesn't decide them.
• A function can have a Taylor series that doesn't converge back to it; the
remainder $R_n\to0$ is what guarantees equality.
• Reuse known series (substitute, differentiate, integrate) instead of computing
every derivative.
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.
Find the radius of convergence of $\displaystyle\sum\frac{x^n}{n!}$.
Use a known series to get the Maclaurin series of $e^{-x^2}$.
Estimate $\displaystyle\int_0^1 e^{-x^2}\,dx$ using the first three terms.
Find the interval of convergence of $\displaystyle\sum\frac{(x-2)^n}{n\, 3^n}$.
Find the Maclaurin series for $\cos x$.
Write $\dfrac{1}{2-x}$ as a power series centred at $0$.
Find the interval of convergence of $\displaystyle\sum_{n=1}^\infty\frac{n^n}{42^n}x^n$.
Estimate $\sin 1$ with error less than $0.001$.
Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:
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.
The spine is: convergence interval first, algebraic manipulation second, Taylor approximation third, and error bound last.
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.
Find the radius of convergence of $\sum n x^n$.
Check convergence of $\sum n x^n$ at $x=1$ and $x=-1$.
Write the degree-3 Taylor polynomial for $e^x$ at $0$.
Give a simple bound for the error in approximating $e^{0.1}$ by $1+0.1+0.1^2/2$.