Practice Problems – Power Series & Taylor Series – AP Calculus BC

Below are 20 practice problems covering all topics from Module 07: Power Series & Taylor/Maclaurin Series (BC). Topics include: radius/interval of convergence, finding Taylor/Maclaurin series, Taylor polynomials, Lagrange error bound, and series manipulations. Each problem includes a hidden solution — click the button to reveal the step-by-step answer.

Problem 01 Power Series

Find the radius of convergence for $\sum_{n=0}^\infty \frac{x^n}{n!}$.

Solution:
Ratio Test: $\lim_{n\to\infty} \frac{x^{n+1}/(n+1)!}{x^n/n!} = \lim_{n\to\infty} \frac{|x|}{n+1} = 0$ for all $x$. So $R = \infty$.

Problem 02 Power Series

Find the interval of convergence for $\sum_{n=1}^\infty \frac{(x-2)^n}{n}$.

Solution:
Ratio Test: $\lim \frac{|x-2|^{n+1}/(n+1)}{|x-2|^n/n} = |x-2|$. Converges when $|x-2| < 1$ → $1 < x < 3$. At $x=3$: $\sum 1/n$ diverges; at $x=1$: $\sum (-1)^n/n$ converges. Interval: $[1, 3)$.

Problem 03 Maclaurin Series

Find the Maclaurin series for $f(x) = e^x$.

Solution:
$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

Problem 04 Maclaurin Series

Find the Maclaurin series for $f(x) = \sin x$.

Solution:
$\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

Problem 05 Taylor Series

Find the Taylor series for $f(x) = \ln x$ centered at $a = 1$.

Solution:
$\ln x = \sum_{n=1}^\infty \frac{(-1)^{n-1} (x-1)^n}{n} = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \cdots$

Problem 06 Series Manipulation

Find the Maclaurin series for $f(x) = e^{2x}$.

Solution:
Substitute $u = 2x$ into $e^u = \sum \frac{u^n}{n!}$: $e^{2x} = \sum_{n=0}^\infty \frac{2^n x^n}{n!}$.

Problem 07 Series Manipulation

Find the Maclaurin series for $f(x) = \sin(x^2)$.

Solution:
$\sin u = \sum \frac{(-1)^n u^{2n+1}}{(2n+1)!}$, let $u = x^2$: $\sin(x^2) = \sum_{n=0}^\infty \frac{(-1)^n x^{4n+2}}{(2n+1)!}$.

Problem 08 Series Manipulation

Find the Maclaurin series for $f(x) = \frac{1}{1+3x}$.

Solution:
$\frac{1}{1-u} = \sum u^n$, let $u = -3x$: $\frac{1}{1+3x} = \sum_{n=0}^\infty (-3)^n x^n = \sum_{n=0}^\infty (-1)^n 3^n x^n$.

Problem 09 Taylor Polynomial

Find the 3rd degree Taylor polynomial for $f(x) = e^x$ centered at $a=0$.

Solution:
$P_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$.

Problem 10 Error Bound

Find the maximum error when using $P_3(x)$ for $e^x$ to approximate $e^{0.2}$.

Solution:
$f^{(4)}(t)=e^t$, $M = e^{0.2} \approx 1.2214$ on $[0,0.2]$. $|R_3| \le \frac{M}{4!}(0.2)^4 = \frac{1.2214}{24} \cdot 0.0016 = 0.0000814$.

Problem 11 Integration of Series

Use the series for $\frac{1}{1+x}$ to find the series for $\ln(1+x)$.

Solution:
$\frac{1}{1+x} = \sum_{n=0}^\infty (-1)^n x^n$, integrate: $\ln(1+x) = \sum_{n=0}^\infty \frac{(-1)^n x^{n+1}}{n+1} = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^n}{n}$.

Problem 12 Differentiation of Series

Use the series for $\frac{1}{1-x}$ to find the series for $\frac{1}{(1-x)^2}$.

Solution:
$\frac{1}{1-x} = \sum_{n=0}^\infty x^n$. Differentiate: $\frac{1}{(1-x)^2} = \sum_{n=1}^\infty n x^{n-1} = \sum_{n=0}^\infty (n+1) x^n$.

Problem 13 Integration of Series

Find the Maclaurin series for $\arctan x$ by integrating $\frac{1}{1+x^2}$.

Solution:
$\frac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^n x^{2n}$. Integrate: $\arctan x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1}$.

Problem 14 Error Bound

How many terms of the Maclaurin series for $e^x$ are needed to approximate $e^{0.5}$ with error less than $0.0001$?

Solution:
$|R_n| \le \frac{e^{0.5}}{(n+1)!}(0.5)^{n+1}$. Try $n=4$: error $\approx 0.00043 > 0.0001$. $n=5$: error $\approx 0.000036 < 0.0001$. So $n=5$ (degree 5 polynomial).

Problem 15 Power Series

Find the radius of convergence for $\sum_{n=0}^\infty n! x^n$.

Solution:
Ratio Test: $\lim \frac{(n+1)! |x|^{n+1}}{n! |x|^n} = \lim (n+1)|x| = \infty$ for $x \neq 0$. Converges only at $x=0$, so $R=0$.

Problem 16 Series Manipulation

Find the Maclaurin series for $f(x) = \cos(2x)$.

Solution:
$\cos u = \sum_{n=0}^\infty \frac{(-1)^n u^{2n}}{(2n)!}$, let $u=2x$: $\cos(2x) = \sum_{n=0}^\infty \frac{(-1)^n (2x)^{2n}}{(2n)!} = \sum_{n=0}^\infty \frac{(-1)^n 4^n x^{2n}}{(2n)!}$.

Problem 17 Series Manipulation

Find the Maclaurin series for $f(x) = \frac{x}{1-x}$.

Solution:
$\frac{1}{1-x} = \sum_{n=0}^\infty x^n$, multiply by $x$: $\frac{x}{1-x} = \sum_{n=0}^\infty x^{n+1} = \sum_{n=1}^\infty x^n$.

Problem 18 Taylor Series

Find the Taylor series for $f(x) = \sin x$ centered at $a = \pi/2$.

Solution:
$f(\pi/2)=1$, $f'(\pi/2)=0$, $f''(\pi/2)=-1$, $f'''(\pi/2)=0$, etc. $\sin x = 1 - \frac{(x-\pi/2)^2}{2!} + \frac{(x-\pi/2)^4}{4!} - \cdots = \sum_{n=0}^\infty \frac{(-1)^n (x-\pi/2)^{2n}}{(2n)!}$.

Problem 19 Error Bound

Find the maximum error when using $P_3(x) = x - \frac{x^3}{6}$ to approximate $\sin(0.4)$.

Solution:
$f^{(4)}(t) = \sin t$, $|f^{(4)}(t)| \le 1$, so $M=1$. $|R_3(0.4)| \le \frac{1}{4!}(0.4)^4 = \frac{1}{24} \cdot 0.0256 = 0.001067$.

Problem 20 Power Series

Find a power series representation for $f(x) = \frac{1}{2-x}$ and state the radius of convergence.

Solution:
$\frac{1}{2-x} = \frac{1}{2} \cdot \frac{1}{1 - x/2} = \frac{1}{2} \sum_{n=0}^\infty \left(\frac{x}{2}\right)^n = \sum_{n=0}^\infty \frac{x^n}{2^{n+1}}$. Converges when $|x/2| < 1$ → $|x| < 2$, so $R=2$.

← Back to Module Page

📝 Practice Problems