40A05-IntegralTest.tex

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\begin{document}
Consider a sequence $(a_n)=\{a_0,a_1,a_2,a_3,\ldots\}$ 
and given $M\in \R$ consider any monotonically nonincreasing function $f:[M,+\infty)\to \R$ 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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\end{document}