Practice Series in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The result of adding all the terms of a sequence together, either finitely or infinitely many terms.

Add up all the terms: $a_1 + a_2 + a_3 + \ldots$ — an infinite series can still have a finite sum if terms shrink fast enough.

Showing a random 20 of 50 problems.

Example 1

medium

Find the sum of the arithmetic series

\sum_{n=1}^{10}(3n+1)

Example 2

easy

Does

\sum_{n=1}^{\infty} n

(i.e.

1+2+3+\cdots

) converge?

Example 3

medium

Determine whether

\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}

converges.

Example 4

medium

Apply the

p

-series test to

\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}

Example 5

easy

What is the difference between

a_n

and

S_n

for a series?

Example 6

easy

Find the sum of the infinite geometric series

1 + \frac{1}{3} + \frac{1}{9} + \cdots

Example 7

easy

What is sigma notation for

\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots

Example 8

medium

Use

\sum_{n=1}^{N} n^2=\frac{N(N+1)(2N+1)}{6}

to find

\sum_{n=1}^{10} n^2

Example 9

medium

Use the divergence test on

\sum_{n=1}^{\infty} \frac{n}{2n+1}

Example 10

easy

Find the 3rd partial sum

S_3

of the series

1+2+3+4+\cdots

Example 11

easy

Find the sum

\sum_{n=1}^{4} (2n - 1)

Example 12

medium

Test

\sum_{n=1}^{\infty} \frac{1}{n(n+1)}

for convergence by finding

S_N

explicitly (telescoping).

Example 13

hard

\sum_{n=1}^{\infty} \frac{n^2}{n^3 + 1}

convergent or divergent?

Example 14

hard

Determine convergence of

\sum_{n=1}^{\infty} \frac{n}{2^n}

and find its sum.

Example 15

medium

Find the sum of the geometric series

\sum_{n=1}^{\infty} \frac{3^n}{4^{n+1}}

Example 16

medium

A series has partial sums

S_n=\frac{n}{n+1}

. What does the series converge to?

Example 17

easy

Does

\sum_{n=1}^{\infty} (-1)^n

converge?

Example 18

easy

Find the second partial sum

S_2

\sum_{n=1}^{\infty} 1/n^2

Example 19

medium

Express the repeating decimal

0.\overline{27}

as a fraction using a geometric series.

Example 20

easy

Does the geometric series

1+\frac12+\frac14+\cdots

converge? To what?

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Related Concepts

Sequence Convergence and Divergence