Calculus

Infinite Sequences and Series

Infinite Sequences and Series Logo

The Integral Test

  • 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

    \[
    {\sum\limits_{n = 1}^\infty {f\left( n \right)} }
    = {f\left( 1 \right) + f\left( 2 \right) }+{ f\left( 3 \right) + \ldots }+{ f\left( n \right) + \ldots }
    \]

    cconverges 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 a problem to see the 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