Use the series for cos x to find the first four nonzero terms of the series for -2x^3cos (x^2)

-2x^3 + x^7 - 1/12x^11 + 1/360x^15 + ×s

Use the series for sin x to find the first four nonzero terms of the series for xsin (3x)

3x^2 - 9/2x^4 + 81/40x^6 - 243/560x^8 + ×s

Use the series for ln (1 + x) to find the first four nonzero terms of the series for 2x^2ln (1 + 2x)

4x^3 - 4x^4 + 16/3x^5 - 8x^6 + ×s

Use the series for ln (1 + x) to find the first four nonzero terms of the series for -xln (1 - x^2)

x^3 + 1/2x^5 + 1/3x^7 + 1/4x^9 + ×s

Use the series for ln (1 + x) to find the first four nonzero terms of the series for ln (1 - x^2)

-x^2 - 1/2x^4 - 1/3x^6 - 1/4x^8 + ×s

Use the series for sin x to find the first four nonzero terms of the series for 3x^2sin (2x)

6x^3 - 4x^5 + 4/5x^7 - 8/105x^9 + ×s

Use the series for (1)/(1 - x) to find the first four nonzero terms of the series for -(2x^2)/(1 + x)

-2x^2 + 2x^3 - 2x^4 + 2x^5 + ×s

Use the series for ln (1 + x) to find the first four nonzero terms of the series for -2x^2ln (1 + 4x^3)

-8x^5 + 16x^8 - 128/3x^11 + 128x^14 + ×s

Use the series for ln (1 + x) to find the first four nonzero terms of the series for -2ln (1 + 4x)

-8x + 16x^2 - 128/3x^3 + 128x^4 + ×s

Use the series for (1)/(1 - x) to find the first four nonzero terms of the series for -(2)/(1 - 3x^3)

-2 - 6x^3 - 18x^6 - 54x^9 + ×s

Use the series for cos x to find the first four nonzero terms of the series for -cos (3x^2)

-1 + 9/2x^4 - 27/8x^8 + 81/80x^12 + ×s

Use the series for ln (1 + x) to find the first four nonzero terms of the series for -2ln (1 - x)

2x + x^2 + 2/3x^3 + 1/2x^4 + ×s

Use the series for cos x to find the first four nonzero terms of the series for xcos (x^2)

x - 1/2x^5 + 1/24x^9 - 1/720x^13 + ×s

Use the series for ln (1 + x) to find the first four nonzero terms of the series for -2x^3ln (1 - x^3)

2x^6 + x^9 + 2/3x^12 + 1/2x^15 + ×s

Worksheets · Calculus

Manipulating power series worksheet

Every series here comes from one you already know: 1/(1 - x), eˣ, sin x, cos x or ln(1 + x). Substitute for x, multiply by a power of x, differentiate or integrate term by term, or add and multiply two series.

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 Manipulating Power Series Date Period Give the first four nonzero terms of each series. Use the series for $\cos x$ to find the first four nonzero terms of the series for $- 2 x^{3} \cos ⁣ (x^{2})$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $- 2 x e^{3 x^{2}}$ Use the series for $\sin x$ to find the first four nonzero terms of the series for $x \sin (3 x)$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x e^{- x}$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $2 x^{2} \ln (1 + 2 x)$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- x \ln ⁣ (1 - x^{2})$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $\ln ⁣ (1 - x^{2})$ Use the series for $\sin x$ to find the first four nonzero terms of the series for $3 x^{2} \sin (2 x)$ Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2 x^{2}}{1 + x}$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{5 x^{2}}$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{2} \ln ⁣ (1 + 4 x^{3})$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 + 4 x)$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x^{3} e^{5 x^{2}}$ Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2}{1 - 3 x^{3}}$ Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $\frac{x}{1 - 2 x}$ Use the series for $\cos x$ to find the first four nonzero terms of the series for $- \cos ⁣ (3 x^{2})$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 - x)$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{- x^{2}}$ Use the series for $\cos x$ to find the first four nonzero terms of the series for $x \cos ⁣ (x^{2})$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{3} \ln ⁣ (1 - x^{3})$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Manipulating Power Series

Date Period

Give the first four nonzero terms of each series.

Use the series for $\cos x$ to find the first four nonzero terms of the series for $- 2 x^{3} \cos ⁣ (x^{2})$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $- 2 x e^{3 x^{2}}$
Use the series for $\sin x$ to find the first four nonzero terms of the series for $x \sin (3 x)$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x e^{- x}$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $2 x^{2} \ln (1 + 2 x)$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- x \ln ⁣ (1 - x^{2})$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $\ln ⁣ (1 - x^{2})$
Use the series for $\sin x$ to find the first four nonzero terms of the series for $3 x^{2} \sin (2 x)$
Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2 x^{2}}{1 + x}$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{5 x^{2}}$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{2} \ln ⁣ (1 + 4 x^{3})$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 + 4 x)$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x^{3} e^{5 x^{2}}$
Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2}{1 - 3 x^{3}}$
Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $\frac{x}{1 - 2 x}$
Use the series for $\cos x$ to find the first four nonzero terms of the series for $- \cos ⁣ (3 x^{2})$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 - x)$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{- x^{2}}$
Use the series for $\cos x$ to find the first four nonzero terms of the series for $x \cos ⁣ (x^{2})$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{3} \ln ⁣ (1 - x^{3})$



Sigma Prep · sigmaprep.io/worksheets Name Manipulating Power SeriesAnswer keyVersion 1 Date Period Give the first four nonzero terms of each series. Use the series for $\cos x$ to find the first four nonzero terms of the series for $- 2 x^{3} \cos ⁣ (x^{2})$ $- 2 x^{3} + x^{7} - \frac{1}{12} x^{11} + \frac{1}{360} x^{15} + \dots$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $- 2 x e^{3 x^{2}}$ $- 2 x - 6 x^{3} - 9 x^{5} - 9 x^{7} + \dots$ Use the series for $\sin x$ to find the first four nonzero terms of the series for $x \sin (3 x)$ $3 x^{2} - \frac{9}{2} x^{4} + \frac{81}{40} x^{6} - \frac{243}{560} x^{8} + \dots$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x e^{- x}$ $3 x - 3 x^{2} + \frac{3}{2} x^{3} - \frac{1}{2} x^{4} + \dots$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $2 x^{2} \ln (1 + 2 x)$ $4 x^{3} - 4 x^{4} + \frac{16}{3} x^{5} - 8 x^{6} + \dots$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- x \ln ⁣ (1 - x^{2})$ $x^{3} + \frac{1}{2} x^{5} + \frac{1}{3} x^{7} + \frac{1}{4} x^{9} + \dots$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $\ln ⁣ (1 - x^{2})$ $- x^{2} - \frac{1}{2} x^{4} - \frac{1}{3} x^{6} - \frac{1}{4} x^{8} + \dots$ Use the series for $\sin x$ to find the first four nonzero terms of the series for $3 x^{2} \sin (2 x)$ $6 x^{3} - 4 x^{5} + \frac{4}{5} x^{7} - \frac{8}{105} x^{9} + \dots$ Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2 x^{2}}{1 + x}$ $- 2 x^{2} + 2 x^{3} - 2 x^{4} + 2 x^{5} + \dots$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{5 x^{2}}$ $x + 5 x^{3} + \frac{25}{2} x^{5} + \frac{125}{6} x^{7} + \dots$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{2} \ln ⁣ (1 + 4 x^{3})$ $- 8 x^{5} + 16 x^{8} - \frac{128}{3} x^{11} + 128 x^{14} + \dots$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 + 4 x)$ $- 8 x + 16 x^{2} - \frac{128}{3} x^{3} + 128 x^{4} + \dots$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x^{3} e^{5 x^{2}}$ $3 x^{3} + 15 x^{5} + \frac{75}{2} x^{7} + \frac{125}{2} x^{9} + \dots$ Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2}{1 - 3 x^{3}}$ $- 2 - 6 x^{3} - 18 x^{6} - 54 x^{9} + \dots$ Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $\frac{x}{1 - 2 x}$ $x + 2 x^{2} + 4 x^{3} + 8 x^{4} + \dots$ Use the series for $\cos x$ to find the first four nonzero terms of the series for $- \cos ⁣ (3 x^{2})$ $- 1 + \frac{9}{2} x^{4} - \frac{27}{8} x^{8} + \frac{81}{80} x^{12} + \dots$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 - x)$ $2 x + x^{2} + \frac{2}{3} x^{3} + \frac{1}{2} x^{4} + \dots$ Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{- x^{2}}$ $x - x^{3} + \frac{1}{2} x^{5} - \frac{1}{6} x^{7} + \dots$ Use the series for $\cos x$ to find the first four nonzero terms of the series for $x \cos ⁣ (x^{2})$ $x - \frac{1}{2} x^{5} + \frac{1}{24} x^{9} - \frac{1}{720} x^{13} + \dots$ Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{3} \ln ⁣ (1 - x^{3})$ $2 x^{6} + x^{9} + \frac{2}{3} x^{12} + \frac{1}{2} x^{15} + \dots$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Manipulating Power SeriesAnswer keyVersion 1

Date Period

Give the first four nonzero terms of each series.

Use the series for $\cos x$ to find the first four nonzero terms of the series for $- 2 x^{3} \cos ⁣ (x^{2})$ $- 2 x^{3} + x^{7} - \frac{1}{12} x^{11} + \frac{1}{360} x^{15} + \dots$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $- 2 x e^{3 x^{2}}$ $- 2 x - 6 x^{3} - 9 x^{5} - 9 x^{7} + \dots$
Use the series for $\sin x$ to find the first four nonzero terms of the series for $x \sin (3 x)$ $3 x^{2} - \frac{9}{2} x^{4} + \frac{81}{40} x^{6} - \frac{243}{560} x^{8} + \dots$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x e^{- x}$ $3 x - 3 x^{2} + \frac{3}{2} x^{3} - \frac{1}{2} x^{4} + \dots$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $2 x^{2} \ln (1 + 2 x)$ $4 x^{3} - 4 x^{4} + \frac{16}{3} x^{5} - 8 x^{6} + \dots$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- x \ln ⁣ (1 - x^{2})$ $x^{3} + \frac{1}{2} x^{5} + \frac{1}{3} x^{7} + \frac{1}{4} x^{9} + \dots$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $\ln ⁣ (1 - x^{2})$ $- x^{2} - \frac{1}{2} x^{4} - \frac{1}{3} x^{6} - \frac{1}{4} x^{8} + \dots$
Use the series for $\sin x$ to find the first four nonzero terms of the series for $3 x^{2} \sin (2 x)$ $6 x^{3} - 4 x^{5} + \frac{4}{5} x^{7} - \frac{8}{105} x^{9} + \dots$
Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2 x^{2}}{1 + x}$ $- 2 x^{2} + 2 x^{3} - 2 x^{4} + 2 x^{5} + \dots$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{5 x^{2}}$ $x + 5 x^{3} + \frac{25}{2} x^{5} + \frac{125}{6} x^{7} + \dots$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{2} \ln ⁣ (1 + 4 x^{3})$ $- 8 x^{5} + 16 x^{8} - \frac{128}{3} x^{11} + 128 x^{14} + \dots$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 + 4 x)$ $- 8 x + 16 x^{2} - \frac{128}{3} x^{3} + 128 x^{4} + \dots$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $3 x^{3} e^{5 x^{2}}$ $3 x^{3} + 15 x^{5} + \frac{75}{2} x^{7} + \frac{125}{2} x^{9} + \dots$
Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $- \frac{2}{1 - 3 x^{3}}$ $- 2 - 6 x^{3} - 18 x^{6} - 54 x^{9} + \dots$
Use the series for $\frac{1}{1 - x}$ to find the first four nonzero terms of the series for $\frac{x}{1 - 2 x}$ $x + 2 x^{2} + 4 x^{3} + 8 x^{4} + \dots$
Use the series for $\cos x$ to find the first four nonzero terms of the series for $- \cos ⁣ (3 x^{2})$ $- 1 + \frac{9}{2} x^{4} - \frac{27}{8} x^{8} + \frac{81}{80} x^{12} + \dots$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 \ln (1 - x)$ $2 x + x^{2} + \frac{2}{3} x^{3} + \frac{1}{2} x^{4} + \dots$
Use the series for $e^{x}$ to find the first four nonzero terms of the series for $x e^{- x^{2}}$ $x - x^{3} + \frac{1}{2} x^{5} - \frac{1}{6} x^{7} + \dots$
Use the series for $\cos x$ to find the first four nonzero terms of the series for $x \cos ⁣ (x^{2})$ $x - \frac{1}{2} x^{5} + \frac{1}{24} x^{9} - \frac{1}{720} x^{13} + \dots$
Use the series for $\ln (1 + x)$ to find the first four nonzero terms of the series for $- 2 x^{3} \ln ⁣ (1 - x^{3})$ $2 x^{6} + x^{9} + \frac{2}{3} x^{12} + \frac{1}{2} x^{15} + \dots$

How to do these

Use the series for $e^{x}$ to find the first four nonzero terms of the series for $e^{x^{2}}$

eᵘ = 1 + u + u²/2 + u³/6 + ...
Replace u with x².

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

Where students go wrong

Squaring the series for eˣ to get the series for e^(x²). Those are different functions: e^(x²) comes from putting x² in for x.

Also called power series from known series or substitution in power series.

Questions about these worksheets

Yes. Every manipulating 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.

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.

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.

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.

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.

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 manipulating 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.

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.

Squaring the series for eˣ to get the series for e^(x²). Those are different functions: e^(x²) comes from putting x² in for x.

Manipulating power series worksheet

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How to do these

Where students go wrong

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