Worksheet · Calculus II · Unit 4B

Taylor Series Worksheet with Answers and Error Practice

Twenty problems on known series, transformations, coefficients, intervals, approximations, and error.

20 problems75 min estimated timeIntermediate to advanced progression

What is included

Practice Maclaurin and Taylor series with complete solutions, interval checks, and error estimates.

Skills assessed

series construction
known-series transformations
intervals of convergence
remainder estimates

Prerequisites

derivatives
infinite series

Taylor Series Worksheet instructional sequence — Twenty problems on known series, transformations, coefficients, intervals, approximations, and error. The numbered labels and written sequence preserve meaning without relying on color.

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Find the degree-3 Maclaurin polynomial for e^x.
Find the degree-5 Maclaurin polynomial for sin x.
Find the degree-4 Maclaurin polynomial for cos x.
Write the first four nonzero terms of ln(1+x).
Expand 1/(1-x) as a power series.
Expand 1/(1+x^2) as a power series.
Expand 1/(1-3x) as a power series.
Find a power series for 1/(1-x)^2.
Find a power series for arctan x by integrating 1/(1+x²), and state its interval of convergence.
Find the degree-2 Taylor polynomial for ln x centered at 1.
Write the Taylor series for e^x centered at a.
If f^(n)(0)=2^n, find the Maclaurin series.
Use T_3 for e^x to approximate e^{0.1}.
Use x-x³/6 to approximate sin(0.2).
Find the radius of $\sum_{n=1}^{\infty} n x^n/4^n$ .
Find the interval of convergence of $\sum_{n=1}^{\infty} x^n/n$ .
Find the interval of convergence of $\sum_{n=1}^{\infty} (x-2)^n/(n3^n)$ .
Find R for $\sum_{n=0}^{\infty} n!(x+1)^n/(2n)!$ .
Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.
How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

01

Problem 1: Find the degree-3 Maclaurin polynomial for e^x.

Answer: $1+x+x^2/2+x^3/6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Use $e^x=\sum_{n=0}^{\infty}x^n/n!$ .
Retaining degrees 0 through 3 gives $1+x+x^2/2+x^3/6$ .

02

Problem 2: Find the degree-5 Maclaurin polynomial for sin x.

Answer: $x-x^3/6+x^5/120$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Use $\sin x=\sum_{n=0}^{\infty}(-1)^nx^{2n+1}/(2n+1)!$ .
Retaining terms through degree 5 gives $x-x^3/6+x^5/120$ .

03

Problem 3: Find the degree-4 Maclaurin polynomial for cos x.

Answer: $1-x^2/2+x^4/24$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Use $\cos x=\sum_{n=0}^{\infty}(-1)^nx^{2n}/(2n)!$ .
Retaining terms through degree 4 gives $1-x^2/2+x^4/24$ .

04

Problem 4: Write the first four nonzero terms of ln(1+x).

Answer: $x-x^2/2+x^3/3-x^4/4$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Integrate the geometric series for $1/(1+x)$ term by term.
The first four nonzero terms are $x-x^2/2+x^3/3-x^4/4$ .

05

Problem 5: Expand 1/(1-x) as a power series.

Answer: $\sum_{n=0}^{\infty}x^n,\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

The geometric identity $1/(1-r)=\sum_{n=0}^{\infty}r^n$ requires $|r|<1$ .
Set $r=x$ to obtain $\sum_{n=0}^{\infty}x^n$ for $|x|<1$ .
Therefore the result is $\sum_{n=0}^{\infty}x^n,\ |x|<1$ .

06

Problem 6: Expand 1/(1+x^2) as a power series.

Answer: $\sum_{n=0}^{\infty}(-1)^nx^{2n},\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Set $r=-x^2$ in the geometric series.
This gives $\sum_{n=0}^{\infty}(-1)^nx^{2n}$ , with $|x|<1$ .
Therefore the result is $\sum_{n=0}^{\infty}(-1)^nx^{2n},\ |x|<1$ .

07

Problem 7: Expand 1/(1-3x) as a power series.

Answer: $\sum_{n=0}^{\infty}3^nx^n,\ |x|<1/3$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Set $r=3x$ in the geometric series.
The condition $|3x|<1$ gives $|x|<1/3$ .
Therefore the result is $\sum_{n=0}^{\infty}3^nx^n,\ |x|<1/3$ .

08

Problem 8: Find a power series for 1/(1-x)^2.

Answer: $\sum_{n=1}^{\infty}nx^{n-1},\ |x|<1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Differentiate $1/(1-x)=\sum_{n=0}^{\infty}x^n$ term by term.
This gives $1/(1-x)^2=\sum_{n=1}^{\infty}nx^{n-1}$ for $|x|<1$ .
Therefore the result is $\sum_{n=1}^{\infty}nx^{n-1},\ |x|<1$ .

09

Problem 9: Find a power series for arctan x by integrating 1/(1+x²), and state its interval of convergence.

Answer: $\sum_{n=0}^{\infty}(-1)^n x^{2n+1}/(2n+1),\ -1\le x\le1$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Integrate $1/(1+x^2)=\sum_{n=0}^{\infty}(-1)^nx^{2n}$ from 0 to $x$ .
The result is $\sum_{n=0}^{\infty}(-1)^nx^{2n+1}/(2n+1)$ ; direct endpoint tests include both $x=-1$ and $x=1$ .
Therefore the result is $\sum_{n=0}^{\infty}(-1)^n x^{2n+1}/(2n+1),\ -1\le x\le1$ .

10

Problem 10: Find the degree-2 Taylor polynomial for ln x centered at 1.

Answer: $(x-1)-(x-1)^2/2$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

For $f(x)=\ln x$ , $f(1)=0$ , $f'(1)=1$ , and $f''(1)=-1$ .
Substitution in the degree-2 Taylor formula gives $(x-1)-(x-1)^2/2$ .

11

Problem 11: Write the Taylor series for e^x centered at a.

Answer: $e^a\sum_{n=0}^{\infty}(x-a)^n/n!$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Every derivative of $e^x$ equals $e^x$ , so $f^{(n)}(a)=e^a$ .
The Taylor formula gives $e^a\sum_{n=0}^{\infty}(x-a)^n/n!$ , convergent for every real $x$ .

12

Problem 12: If f^(n)(0)=2^n, find the Maclaurin series.

Answer: $\sum_{n=0}^{\infty}2^nx^n/n!=e^{2x}$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

The Maclaurin coefficient is $f^{(n)}(0)/n!=2^n/n!$ .
Thus the series is $\sum_{n=0}^{\infty}2^nx^n/n!=e^{2x}$ .

13

Problem 13: Use T_3 for e^x to approximate e^{0.1}.

Answer: $1.105166\overline6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Evaluate $T_3(0.1)=1+0.1+0.1^2/2+0.1^3/6$ .
The result is $1.105166\overline6$ .

14

Problem 14: Use x-x³/6 to approximate sin(0.2).

Answer: $0.198666\overline6$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Evaluate $0.2-(0.2)^3/6$ .
The result is $0.198666\overline6$ .

15

Problem 15: Find the radius of $\sum_{n=1}^{\infty} n x^n/4^n$ .

Answer: $R=4$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

The ratio of successive absolute terms approaches $|x|/4$ .
Convergence requires $|x|<4$ , so $R=4$ .

16

Problem 16: Find the interval of convergence of $\sum_{n=1}^{\infty} x^n/n$ .

Answer: $[-1,1)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

The radius is 1. At $x=1$ the harmonic series diverges; at $x=-1$ the alternating harmonic series converges.
Therefore the interval is $[-1,1)$ .

17

Problem 17: Find the interval of convergence of $\sum_{n=1}^{\infty} (x-2)^n/(n3^n)$ .

Answer: $[-1,5)$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

The ratio test gives $|x-2|<3$ , hence $-1<x<5$ .
At $x=-1$ the series is alternating harmonic and converges; at $x=5$ it is harmonic and diverges, giving $[-1,5)$ .

18

Problem 18: Find R for $\sum_{n=0}^{\infty} n!(x+1)^n/(2n)!$ .

Answer: $R=\infty$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

The coefficient ratio satisfies $|a_n/a_{n+1}|=(2n+2)(2n+1)/(n+1)=2(2n+1)$ .
This tends to infinity, so $R=\infty$ .

19

Problem 19: Bound the error of the degree-3 Maclaurin approximation to e^{0.2}.

Answer: $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

Taylor's theorem gives $|R_3|\le e^{0.2}(0.2)^4/4!$ on $[0,0.2]$ .
The bound is approximately $0.00008143<0.000082$ .
Therefore the result is $|R_3|\le e^{0.2}(0.2)^4/4!<0.000082$ .

20

Problem 20: How many terms of the alternating arctan series ensure error below 0.001 at x=1/2?

Answer: $4\text{ terms}$

Why this method: Taylor or power-series construction matches the mathematical structure before any algebraic cleanup.

After $N$ terms, the alternating-series error is at most $(1/2)^{2N+1}/(2N+1)$ .
Three terms give about $0.001116>0.001$ , while four give about $0.000217<0.001$ , so four terms are required.
Therefore the result is $4\text{ terms}$ .

Common errors

Dropping factorials in coefficient formulas.
Transforming a series without its convergence interval.
Using an error estimate without checking its hypotheses.