PlanetMath: integral test

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integral test

(Theorem)

Consider a sequence $(a_n)=\{a_0,a_1,a_2,a_3,\ldots\}$ and given $M\in \mathbbmss{R}$ consider any monotonically nonincreasing function $f:[M,+\infty)\to \mathbbmss{R}$ which extends the sequence, i.e.

$\begin{displaymath} f(n) = a_n \qquad \forall n\ge M \end{displaymath}$

An example is

$\begin{displaymath}a_n = 2n\qquad \to\qquad f(x) = 2x\end{displaymath}$

(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

$\begin{displaymath}\sum_{n = 0}^{\infty}a_n\end{displaymath}$

converges.

The integral test states the following.

The series

$\begin{displaymath}\sum_{n = 0}^{\infty}a_n\end{displaymath}$

converges if and only if the integral

$\begin{displaymath}\int_M^\infty f(x)\, dx\end{displaymath}$

is finite.

"integral test" is owned by drini. [ full author list (3) | owner history (3) ]

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