The Integral Test

Given any series, your _primary consideration_ is determining whether the series converges or diverges.

For a series with non-negative terms, convergence is possible only if the __sequence of partial sums is bounded.__

Integral Test

Here's the basic premise of the Integral Test:

Compare the series to an improper integral of the same function formula as $a_n$.
If the area under the curve of the integrand, $f(x)$, is greater than the sequence of partial sums, the series converges.
If the area under the curve of $f(x)$ is less than the sequence of partial sums, the series diverges.

It's that simple.

Let ${a_n}$ be a sequence of positive terms.

Suppose that $a_n = f(n)$, and $f$ is a continuous, positive, decreasing function of $x$ for all $x \ge N$ (an arbitrary positive integer).

Then:
the series $\sum_{n=N}^{\infty} a_n$ and the (improper) integral $\int_N^{\infty} f(x) dx$ both converge or diverge.



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from IPython.display import SVG
SVG(filename="integral_test_visualization.svg")









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