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Consider a sequence $(a_n)=\{a_0,a_1,a_2,a_3,\ldots\}$ and given
 consider any monotonically nonincreasing function
 which extends the sequence, i.e.$$ f(n) = a_n \qquad \forall n\ge M$$
An example is $$a_n = 2n\qquad \to\qquad f(x) = 2x$$ (the former being the sequence $\{0,2,4,6,8,\ldots\}$ and the later the doubling function for any real number.
We are interested on finding out when the summation $$\sum_{n = 0}^{\infty}a_n$$ converges.
The integral test states the following.
The series $$\sum_{n = 0}^{\infty}a_n$$ converges if and only if the integral $$\int_M^\infty f(x)\, dx$$ is finite.
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