Power Series & Taylor Series
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.
- Find the radius and interval of convergence of a power series.
- Write Taylor and Maclaurin series and know the common ones.
- Bound the error with Taylor's remainder.
- Differentiate and integrate power series term by term.
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.
Power-series workflow
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.
Building new series from old ones
Do not compute every Taylor series from derivatives. Most MAT137 problems are faster if you start from a known series and transform it carefully.
- Substitute. Replace $x$ by a new expression, such as $x^2$ or $-x$.
- Differentiate. Differentiate term by term inside the radius of convergence.
- Integrate. Integrate term by term and then determine the constant using a known value.
- Bound error. Use Taylor's theorem or alternating-series estimates to say how far a polynomial approximation may be from the function.
Power series and radius of convergence
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 $[-1,1)$.
Taylor and Maclaurin series
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 $x$), $\dfrac{1}{1-x}=\displaystyle\sum_{n=0}^\infty x^n$ ($|x|<1$), $\ln(1+x)=\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}x^n}{n}$ ($-1<x\le1$).
Taylor's theorem with remainder
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 $\dfrac{(0.1)^4}{4!}\approx4.2\times10^{-6}$.
Term-by-term calculus
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.
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.
Find the radius of convergence of $\displaystyle\sum\frac{x^n}{n!}$.
Answer
Use a known series to get the Maclaurin series of $e^{-x^2}$.
Answer
Estimate $\displaystyle\int_0^1 e^{-x^2}\,dx$ using the first three terms.
Hint
Find the interval of convergence of $\displaystyle\sum\frac{(x-2)^n}{n\, 3^n}$.
Answer
Find the Maclaurin series for $\cos x$.
Answer
Write $\dfrac{1}{2-x}$ as a power series centred at $0$.
Solution
Find the interval of convergence of $\displaystyle\sum_{n=1}^\infty\frac{n^n}{42^n}x^n$.
Answer
Estimate $\sin 1$ with error less than $0.001$.
Solution
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:
- Power series. Find radius and interval of convergence and check endpoints separately.
- Operations. Differentiate and integrate power series inside their radius of convergence.
- Taylor theorem. Use Taylor polynomials with a remainder term, not just formal pattern matching.
- Approximation. Control error using alternating, Lagrange, or geometric remainder estimates.
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.
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.
Find the radius of convergence of $\sum n x^n$.
Guide
Check convergence of $\sum n x^n$ at $x=1$ and $x=-1$.
Guide
Write the degree-3 Taylor polynomial for $e^x$ at $0$.
Guide
Give a simple bound for the error in approximating $e^{0.1}$ by $1+0.1+0.1^2/2$.