integral test

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

$$f(n)=a_n\mathit{\hspace{1em}\hspace{1em}}\forall n\ge M$$

An example is

$$a_n=2n\mathit{\hspace{1em}\hspace{1em}}\to \mathit{\hspace{1em}\hspace{1em}}f(x)=2x$$

(the former being the sequence $\{0,2,4,6,8,\mathrm{\dots }\}$ and the later the doubling function for any real number.

We are interested on finding out when the summation

$$\sum _{n=0}^{\mathrm{\infty }}{a}_n$$

converges.

The integral test states the following.

The series

$$\sum _{n=0}^{\mathrm{\infty }}{a}_n$$

converges if and only if the integral

$${\int }_M^{\mathrm{\infty }}f(x)𝑑x$$

is finite.