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The Integral Test
Picture infinitely many rectangles of width 1 and height $a_n$, so the area of the $n^{th}$ rectangle is $a_n$. Then the series $\displaystyle\sum_{n=1}^\infty a_n$ is equal to the sum of the areas of these infinitely many rectangles. See the graphic examples below.
From our work with improper integrals, you may have seen that the improper integral $\displaystyle\int_1^\infty\frac{1}{x^p}\,dx$
converges if $p>1$, and diverges if $p\le 1$.
By using the integral test, we therefore get our
$p$series test, which is
extremely useful, especially when used to find comparable series
for the comparison tests.
Explanation and examples of the integral test, as well as determining the above integral of $\frac{1}{x^p}$ and the $p$series test are included on the first video. The second video includes detail of the graphical information above.
