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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