Calculus

Infinite Sequences and Series

The Integral Test

Page 1
Problems 1-2
Page 2
Problems 3-6

Let \(f\left( x \right)\) be a function which is continuous, positive, and decreasing for all \(x\) in the range \(\left[ {1, + \infty } \right).\) Then the series
\[
{\sum\limits_{n = 1}^\infty {f\left( n \right)} }
= {f\left( 1 \right) + f\left( 2 \right) }+{ f\left( 3 \right) + \ldots }+{ f\left( n \right) + \ldots }
\] converges if the improper integral \(\int\limits_1^\infty {f\left( x \right)dx}\) converges, and diverges if \(\int\limits_1^\infty {f\left( x \right)dx} \to \infty.\)

Solved Problems

Click on problem description to see solution.

 Example 1

Determine whether the series \(\sum\limits_{n = 1}^\infty {\large\frac{1}{{1 + 10n}}\normalsize}\) converges or diverges.

 Example 2

Show that the \(p\)-series \(\sum\limits_{n = 1}^\infty {\large\frac{1}{{{n^p}}}\normalsize} \) converges for \(p \gt 1.\)

 Example 3

Determine whether the series \(\sum\limits_{n = 1}^\infty {\large\frac{1}{{\left( {n + 1} \right)\ln \left( {n + 1} \right)}}\normalsize}\) converges or diverges.

 Example 4

Investigate the series \(\sum\limits_{n = 1}^\infty {\large\frac{n}{{{n^2} + 1}}}\normalsize \) for convergence.

 Example 5

Determine whether \(\sum\limits_{n = 1}^\infty {\large\frac{{\arctan n}}{{1 + {n^2}}}\normalsize}\) converges or diverges.

 Example 6

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

Example 1.

Determine whether the series \(\sum\limits_{n = 1}^\infty {\large\frac{1}{{1 + 10n}}\normalsize}\) converges or diverges.

Solution.

We use the integral test. Calculate the improper integral
\[
{\int\limits_1^\infty {\frac{{dx}}{{1 + 10x}}} }
= {\lim\limits_{n \to \infty } \int\limits_1^n {\frac{{dx}}{{1 + 10x}}} }
= {\lim\limits_{n \to \infty } \left. {\left[ {\frac{1}{{10}}\ln \left( {1 + 10x} \right)} \right]} \right|_1^n }
= {\frac{1}{{10}}\lim\limits_{n \to \infty } \left[ {\ln \left( {1 + 10n} \right) }\right.}-{\left.{ \ln 11} \right] }={ \infty .}
\] Thus, the given series is divergent.

Example 2.

Show that the \(p\)-series \(\sum\limits_{n = 1}^\infty {\large\frac{1}{{{n^p}}}\normalsize} \) converges for \(p \gt 1.\)

Solution.

We consider the corresponding function \(f\left( x \right) = \large\frac{1}{{{x^p}}}\normalsize\) and apply the integral test. The improper integral is
\[
{\int\limits_1^\infty {\frac{{dx}}{{{x^p}}}} }
= {\lim\limits_{n \to \infty } \int\limits_1^n {\frac{{dx}}{{{x^p}}}} }
= {\lim\limits_{n \to \infty } \int\limits_1^n {{x^{ – p}}dx} }
= {\lim\limits_{n \to \infty } \left. {\left( {\frac{1}{{ – p + 1}}{x^{ – p + 1}}} \right)} \right|_1^n }
= {\frac{1}{{1 – p}}\lim\limits_{n \to \infty } \left. {\left( {\frac{1}{{{x^{p – 1}}}}} \right)} \right|_1^n }
= {\frac{1}{{1 – p}}\lim\limits_{n \to \infty } \left( {\frac{1}{{{n^{p – 1}}}} – 1} \right) }
= {\frac{1}{{p – 1}}.}
\] Hence, the \(p\)-series converges for \(p \gt 1.\)

Page 1
Problems 1-2
Page 2
Problems 3-6