Series Integral Test Calculator With Steps, Online Series Calculator With Steps

Let (fleft( x
ight)) be a function which is continuous, positive, and decreasing for all (x) in the range (left< {1, + infty } ight).) Then the series

<{sumlimits_{n = 1}^infty {fleft( n ight)} }= {fleft( 1 ight) + fleft( 2 ight) }+{ fleft( 3 ight) + ldots }+{ fleft( n ight) + ldots }>

converges if the improper integral (intlimits_1^infty {fleft( x
ight)dx}) converges, and diverges if (intlimits_1^infty {fleft( x
ight)dx} o infty.)

Solved Problems

Click or tap a problem to see the solution.

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Example 1

Determine whether the series (sumlimits_{n = 1}^infty {largefrac{1}{{1 + 10n}}
ormalsize}) converges or diverges.

Example 2

Show that the (p)-series (sumlimits_{n = 1}^infty {largefrac{1}{{{n^p}}}
ormalsize} ) converges for (p gt 1.)

Example 3

Determine whether the series (sumlimits_{n = 1}^infty {largefrac{1}{{left( {n + 1}
ight)ln left( {n + 1}
ight)}}
ormalsize}) converges or diverges.

Example 4

Investigate the series (sumlimits_{n = 1}^infty {largefrac{n}{{{n^2} + 1}}}
ormalsize ) for convergence.

Example 5

Determine whether (sumlimits_{n = 1}^infty {largefrac{{arctan n}}{{1 + {n^2}}}
ormalsize}) converges or diverges.

Example 6

Investigate whether the series (sumlimits_{n = 0}^infty {n{e^{ – n}}}) converges or diverges.

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Example 1.

Determine whether the series (sumlimits_{n = 1}^infty {largefrac{1}{{1 + 10n}}
ormalsize}) converges or diverges.

Solution.

We use the integral test. Calculate the improper integral

<{intlimits_1^infty {frac{{dx}}{{1 + 10x}}} }= {limlimits_{n o infty } intlimits_1^n {frac{{dx}}{{1 + 10x}}} }= {limlimits_{n o infty } left. {left< {frac{1}{{10}}ln left( {1 + 10x} ight)} ight>}
ight|_1^n }= {frac{1}{{10}}limlimits_{n o infty } left< {ln left( {1 + 10n} ight) } ight.}-{left.{ ln 11} ight> }={ infty .}>

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Thus, the given series is divergent.

Example 2.

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Show that the (p)-series (sumlimits_{n = 1}^infty {largefrac{1}{{{n^p}}}
ormalsize} ) converges for (p gt 1.)

Solution.

We consider the corresponding function (fleft( x
ight) = largefrac{1}{{{x^p}}}
ormalsize) and apply the integral test. The improper integral is

<{intlimits_1^infty {frac{{dx}}{{{x^p}}}} }= {limlimits_{n o infty } intlimits_1^n {frac{{dx}}{{{x^p}}}} }= {limlimits_{n o infty } intlimits_1^n {{x^{ – p}}dx} }= {limlimits_{n o infty } left. {left( {frac{1}{{ – p + 1}}{x^{ – p + 1}}} ight)} ight|_1^n }= {frac{1}{{1 – p}}limlimits_{n o infty } left. {left( {frac{1}{{{x^{p – 1}}}}} ight)} ight|_1^n }= {frac{1}{{1 – p}}limlimits_{n o infty } left( {frac{1}{{{n^{p – 1}}}} – 1} ight) }= {frac{1}{{p – 1}}.}>

Hence, the (p)-series converges for (p gt 1.)

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