# The Integral Test

Let f (x) be a function which is continuous, positive, and decreasing for all x in the range [1, +∞). 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 or tap a problem to see the solution.

### Example 1

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

### Example 2

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

### Example 1.

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

Solution.

We use the integral test. Calculate the improper integral

$\int\limits_1^\infty {\frac{{dx}}{{1 + 10x}}} = \lim\limits_{n \to \infty } \int\limits_1^n {\frac{{dx}}{{1 + 10x}}} = \lim\limits_{n \to \infty } \left. {\left[ {\frac{1}{{10}}\ln \left( {1 + 10x} \right)} \right]} \right|_1^n = \frac{1}{{10}}\lim\limits_{n \to \infty } \left[ {\ln \left( {1 + 10n} \right) - \ln 11} \right] = \infty .$

Thus, the given series is divergent.

### Example 2.

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

Solution.

We consider the corresponding function $$f\left( x \right) = \frac{1}{{{x^p}}}$$ 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 $$p \gt 1.$$

See more problems on Page 2.