Write as a power series, and give the interval of convergence: f(x) = (1)/(1 - 5x)

of_n=0^infinity 5^n x^n, (-1/5, 1/5)

Write as a power series, and give the interval of convergence: f(x) = (4)/(1 + 2x)

of_n=0^infinity 4(-2)^n x^n, (-1/2, 1/2)

Write as a power series, and give the interval of convergence: f(x) = -(2)/(1 - 4x)

of_n=0^infinity -2 × 4^n x^n, (-1/4, 1/4)

Write as a power series, and give the interval of convergence: f(x) = -(1)/(1 + 3x)

of_n=0^infinity -(-3)^n x^n, (-1/3, 1/3)

Write as a power series, and give the interval of convergence: f(x) = -(5)/(1 + 4x)

of_n=0^infinity -5(-4)^n x^n, (-1/4, 1/4)

Write as a power series, and give the interval of convergence: f(x) = -(7)/(1 - 3x)

of_n=0^infinity -7 × 3^n x^n, (-1/3, 1/3)

Write as a power series, and give the interval of convergence: f(x) = (5)/(1 + 5x)

of_n=0^infinity 5(-5)^n x^n, (-1/5, 1/5)

Write as a power series, and give the interval of convergence: f(x) = (3)/(1 - 4x)

of_n=0^infinity 3 × 4^n x^n, (-1/4, 1/4)

Write as a power series and give the interval of convergence: f(x) = (4)/(3 - x^2)

sum of_n=0^infinity (4)/(3^n + 1)x^2n; |x| < sqrt(3)

On one grid, graph g(x) = (1)/(1 + x) and its partial sum S_3(x) (the first 3 terms of its series). Where do they agree?

S_3(x) = 1 - x + x^2; close near x = 0: off by 0.083 at x = 0.5 but 0.384 at x = 0.9; apart for |x| ≥ 1, where the series diverges

Worksheets · Calculus

Power series worksheet

Q: Are these power series worksheets free?

Yes. Every power series sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Q: Do the power series worksheets come with an answer key?

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Q: Are these power series worksheets printable?

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Q: What grade are these power series worksheets for?

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Q: Can I get a different version of the same power series worksheet?

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

Q: Is this a free alternative to Kuta Software for power series?

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the power series questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Q: Can I choose how hard the power series questions are?

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Q: What kinds of question are on a power series worksheet?

You pick from 6: a function as a power series, the interval of convergence, the first terms of a maclaurin series, approximate e^a, sin or cos with a partial sum, substitute x^2 into a geometric series and graph a function and a partial sum. Tick as many as you want and set how many of each, or let it spread them evenly.

Q: What do students get wrong with power series?

Stopping at the radius. The ratio test is inconclusive exactly at the endpoints, and each one can go either way: the same radius can give an open, closed or half-open interval.

In Calculus power series are a main topic of the series unit: you write functions as series, find the radius and interval of convergence with the ratio test, and build Maclaurin series from derivatives or by substituting into known series. Endpoints are checked with the earlier series tests.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Power Series Date Period Answer each question. Write as a power series Write as a power series, and give the interval of convergence: $f (x) = \frac{1}{1 - 5 x}$ Write as a power series, and give the interval of convergence: $f (x) = \frac{4}{1 + 2 x}$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{2}{1 - 4 x}$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{1}{1 + 3 x}$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{5}{1 + 4 x}$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{7}{1 - 3 x}$ Write as a power series, and give the interval of convergence: $f (x) = \frac{5}{1 + 5 x}$ Find the interval of convergence Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{3^{n}}$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{3^{n}}$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{x^{n}}{2^{n}}$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x - 2)}^{n}}{5^{n}}$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{6^{n}}$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x + 4)}^{n}}{6^{n}}$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{4^{n}}$ Write the first Maclaurin series terms Write the first four nonzero terms of the Maclaurin series for $\cos (3 x)$ Write the first four nonzero terms of the Maclaurin series for $e^{- 2 x}$ Write the first four nonzero terms of the Maclaurin series for $\sin (3 x)$ Write the first four nonzero terms of the Maclaurin series for $\frac{1}{1 - 2 x}$ Write the first four nonzero terms of the Maclaurin series for $\ln (1 + 3 x)$ Write the first four nonzero terms of the Maclaurin series for $\ln (1 - 2 x)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Power Series

Date Period

Answer each question.

Write as a power series
Write as a power series, and give the interval of convergence:
$f (x) = \frac{1}{1 - 5 x}$
Write as a power series, and give the interval of convergence:
$f (x) = \frac{4}{1 + 2 x}$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{2}{1 - 4 x}$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{1}{1 + 3 x}$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{5}{1 + 4 x}$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{7}{1 - 3 x}$
Write as a power series, and give the interval of convergence:
$f (x) = \frac{5}{1 + 5 x}$

Find the interval of convergence
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{3^{n}}$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{3^{n}}$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{x^{n}}{2^{n}}$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x - 2)}^{n}}{5^{n}}$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{6^{n}}$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x + 4)}^{n}}{6^{n}}$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{4^{n}}$

Write the first Maclaurin series terms
Write the first four nonzero terms of the Maclaurin series for $\cos (3 x)$
Write the first four nonzero terms of the Maclaurin series for $e^{- 2 x}$
Write the first four nonzero terms of the Maclaurin series for $\sin (3 x)$
Write the first four nonzero terms of the Maclaurin series for $\frac{1}{1 - 2 x}$
Write the first four nonzero terms of the Maclaurin series for $\ln (1 + 3 x)$
Write the first four nonzero terms of the Maclaurin series for $\ln (1 - 2 x)$



Sigma Prep · sigmaprep.io/worksheets Name Power SeriesAnswer keyVersion 1 Date Period Answer each question. Write as a power series Write as a power series, and give the interval of convergence: $f (x) = \frac{1}{1 - 5 x}$ $\sum_{n = 0}^{\infty} 5^{n} x^{n}$ , $(- \frac{1}{5}, \frac{1}{5})$ Write as a power series, and give the interval of convergence: $f (x) = \frac{4}{1 + 2 x}$ $\sum_{n = 0}^{\infty} 4 {(- 2)}^{n} x^{n}$ , $(- \frac{1}{2}, \frac{1}{2})$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{2}{1 - 4 x}$ $\sum_{n = 0}^{\infty} - 2 \cdot 4^{n} x^{n}$ , $(- \frac{1}{4}, \frac{1}{4})$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{1}{1 + 3 x}$ $\sum_{n = 0}^{\infty} - {(- 3)}^{n} x^{n}$ , $(- \frac{1}{3}, \frac{1}{3})$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{5}{1 + 4 x}$ $\sum_{n = 0}^{\infty} - 5 {(- 4)}^{n} x^{n}$ , $(- \frac{1}{4}, \frac{1}{4})$ Write as a power series, and give the interval of convergence: $f (x) = - \frac{7}{1 - 3 x}$ $\sum_{n = 0}^{\infty} - 7 \cdot 3^{n} x^{n}$ , $(- \frac{1}{3}, \frac{1}{3})$ Write as a power series, and give the interval of convergence: $f (x) = \frac{5}{1 + 5 x}$ $\sum_{n = 0}^{\infty} 5 {(- 5)}^{n} x^{n}$ , $(- \frac{1}{5}, \frac{1}{5})$ Find the interval of convergence Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{3^{n}}$ $(2, 8)$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{3^{n}}$ $(- 4, 2)$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{x^{n}}{2^{n}}$ $(- 2, 2)$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x - 2)}^{n}}{5^{n}}$ $(- 3, 7)$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{6^{n}}$ $(- 7, 5)$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x + 4)}^{n}}{6^{n}}$ $(- 10, 2)$ Find the interval of convergence: $\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{4^{n}}$ $(1, 9)$ Write the first Maclaurin series terms Write the first four nonzero terms of the Maclaurin series for $\cos (3 x)$ $1 - \frac{9}{2} x^{2} + \frac{27}{8} x^{4} - \frac{81}{80} x^{6}$ Write the first four nonzero terms of the Maclaurin series for $e^{- 2 x}$ $1 - 2 x + 2 x^{2} - \frac{4}{3} x^{3}$ Write the first four nonzero terms of the Maclaurin series for $\sin (3 x)$ $3 x - \frac{9}{2} x^{3} + \frac{81}{40} x^{5} - \frac{243}{560} x^{7}$ Write the first four nonzero terms of the Maclaurin series for $\frac{1}{1 - 2 x}$ $1 + 2 x + 4 x^{2} + 8 x^{3}$ Write the first four nonzero terms of the Maclaurin series for $\ln (1 + 3 x)$ $3 x - \frac{9}{2} x^{2} + 9 x^{3} - \frac{81}{4} x^{4}$ Write the first four nonzero terms of the Maclaurin series for $\ln (1 - 2 x)$ $- 2 x - 2 x^{2} - \frac{8}{3} x^{3} - 4 x^{4}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Power SeriesAnswer keyVersion 1

Date Period

Answer each question.

Write as a power series
Write as a power series, and give the interval of convergence:
$f (x) = \frac{1}{1 - 5 x}$ $\sum_{n = 0}^{\infty} 5^{n} x^{n}$ , $(- \frac{1}{5}, \frac{1}{5})$
Write as a power series, and give the interval of convergence:
$f (x) = \frac{4}{1 + 2 x}$ $\sum_{n = 0}^{\infty} 4 {(- 2)}^{n} x^{n}$ , $(- \frac{1}{2}, \frac{1}{2})$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{2}{1 - 4 x}$ $\sum_{n = 0}^{\infty} - 2 \cdot 4^{n} x^{n}$ , $(- \frac{1}{4}, \frac{1}{4})$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{1}{1 + 3 x}$ $\sum_{n = 0}^{\infty} - {(- 3)}^{n} x^{n}$ , $(- \frac{1}{3}, \frac{1}{3})$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{5}{1 + 4 x}$ $\sum_{n = 0}^{\infty} - 5 {(- 4)}^{n} x^{n}$ , $(- \frac{1}{4}, \frac{1}{4})$
Write as a power series, and give the interval of convergence:
$f (x) = - \frac{7}{1 - 3 x}$ $\sum_{n = 0}^{\infty} - 7 \cdot 3^{n} x^{n}$ , $(- \frac{1}{3}, \frac{1}{3})$
Write as a power series, and give the interval of convergence:
$f (x) = \frac{5}{1 + 5 x}$ $\sum_{n = 0}^{\infty} 5 {(- 5)}^{n} x^{n}$ , $(- \frac{1}{5}, \frac{1}{5})$

Find the interval of convergence
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{3^{n}}$ $(2, 8)$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{3^{n}}$ $(- 4, 2)$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{x^{n}}{2^{n}}$ $(- 2, 2)$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x - 2)}^{n}}{5^{n}}$ $(- 3, 7)$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x + 1)}^{n}}{6^{n}}$ $(- 7, 5)$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x + 4)}^{n}}{6^{n}}$ $(- 10, 2)$
Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{{(x - 5)}^{n}}{4^{n}}$ $(1, 9)$

Write the first Maclaurin series terms
Write the first four nonzero terms of the Maclaurin series for $\cos (3 x)$ $1 - \frac{9}{2} x^{2} + \frac{27}{8} x^{4} - \frac{81}{80} x^{6}$
Write the first four nonzero terms of the Maclaurin series for $e^{- 2 x}$ $1 - 2 x + 2 x^{2} - \frac{4}{3} x^{3}$
Write the first four nonzero terms of the Maclaurin series for $\sin (3 x)$ $3 x - \frac{9}{2} x^{3} + \frac{81}{40} x^{5} - \frac{243}{560} x^{7}$
Write the first four nonzero terms of the Maclaurin series for $\frac{1}{1 - 2 x}$ $1 + 2 x + 4 x^{2} + 8 x^{3}$
Write the first four nonzero terms of the Maclaurin series for $\ln (1 + 3 x)$ $3 x - \frac{9}{2} x^{2} + 9 x^{3} - \frac{81}{4} x^{4}$
Write the first four nonzero terms of the Maclaurin series for $\ln (1 - 2 x)$ $- 2 x - 2 x^{2} - \frac{8}{3} x^{3} - 4 x^{4}$

How to do these

Write the first four nonzero terms of the Maclaurin series for $e^{- x^{2}}$

Start from e^u = 1 + u + u²/2! + u³/3! + ...
Put u = -x².
The terms are 1, -x², x⁴/2 and -x⁶/6.

The answer is $1 - x^{2} + \frac{1}{2} x^{4} - \frac{1}{6} x^{6}$ .

Where students go wrong

Forgetting the factorials. The coefficient of xⁿ in a Maclaurin series is f⁽ⁿ⁾(0)/n!, so leaving out the n! gives terms that are far too large.

Also called power series, interval of convergence, radius of convergence, Maclaurin series or representing functions as power series.

What you can put on this worksheet

A function as a power series: Write as a power series, and give the interval of convergence:
$f (x) = \frac{3}{1 - 4 x}$ → $\sum_{n = 0}^{\infty} 3 \cdot 4^{n} x^{n}$ , $(- \frac{1}{4}, \frac{1}{4})$
The interval of convergence: Find the interval of convergence:
$\sum_{n = 0}^{\infty} \frac{x^{n}}{6^{n}}$ → $(- 6, 6)$
The first terms of a Maclaurin series: Write the first four nonzero terms of the Maclaurin series for $\cos (2 x)$ → $1 - 2 x^{2} + \frac{2}{3} x^{4} - \frac{4}{45} x^{6}$
Approximate e^a, sin or cos with a partial sum: Use the first five terms of the power series to approximate $e^{- 0.2}$ to the nearest thousandth. → $0.819$
Substitute x^2 into a geometric series: Write as a power series and give the interval of convergence:
$f (x) = \frac{4}{3 - x^{2}}$ → $\sum_{n = 0}^{\infty} \frac{4}{3^{n + 1}} x^{2 n}$ ; $∣ x ∣ < \sqrt{3}$
Graph a function and a partial sum: On one grid, graph $g (x) = \frac{1}{1 + x}$ and its partial sum $S_{3} (x)$ (the first 3 terms of its series). Where do they agree? → $S_{3} (x) = 1 - x + x^{2}$ ; close near $x = 0$ : off by $0.083$ at $x = 0.5$ but $0.384$ at $x = 0.9$ ; apart for $∣ x ∣ \geq 1$ , where the series diverges

Questions about these worksheets