# integral test

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.

 $f(n)=a_{n}\qquad\forall n\geq M$

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

 $\sum_{n=0}^{\infty}a_{n}$

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.

Title integral test IntegralTest 2013-03-22 12:27:12 2013-03-22 12:27:12 drini (3) drini (3) 20 drini (3) Theorem msc 40A05 Function Sequence Limit