# Show that an Infinite Series is (absolutely) Convergent

#### NixLUKE

Hello,

New to this topic after missing it at uni due to sickness! Would anyone be able to help me to do this question as im not really sure how to prove that something is convergent for all values of X.

[I do understand how to show that a series converges for a certain value of X]

#### thisuserhas

All major convergence tests are detailed here:
https://en.wikipedia.org/wiki/Convergence_tests
I generally start with the limit test and then move to more 'complicated' tests if it renders an inconclusive result.
Hint - for 1(b)
$$\displaystyle a_{n} = x^{n}/n!$$

#### Archie

For any real $$\displaystyle x$$, pick $$\displaystyle N \in \mathbb N$$ such that $$\displaystyle N > x$$ and break the sum into two parts:

$$\displaystyle \sum \limits_{n=0}^\infty {x^n \over n!} = \sum \limits_{n=0}^{N-1} {x^n \over n!} + \sum \limits_{n=N}^\infty {x^n \over n!} < \sum \limits_{n=0}^{N-1} {x^n \over n!} + {x^N \over N!} \sum \limits_{n=0}^\infty {x^n \over "^n}$$​

The first term is a finite sum and the summation in the second term is a convergent geometric series.

1 person

#### Archie

Oops, typo:
For any real $$\displaystyle x$$, pick $$\displaystyle N \in \mathbb N$$ such that $$\displaystyle N > x$$ and break the sum into two parts:

$$\displaystyle \sum \limits_{n=0}^\infty {x^n \over n!} = \sum \limits_{n=0}^{N-1} {x^n \over n!} + \sum \limits_{n=N}^\infty {x^n \over n!} < \sum \limits_{n=0}^{N-1} {x^n \over n!} + {x^N \over N!} \sum \limits_{n=0}^\infty {x^n \over N^n}$$​

The first term is a finite sum and the summation in the second term is a convergent geometric series.

#### skeeter

MHF Helper
ratio test to determine the interval of convergence ...

$\displaystyle \lim_{n \to \infty} \bigg|\dfrac{a_{n+1}}{a_n}\bigg| < 1$

$\displaystyle \lim_{n \to \infty} \bigg|\dfrac{x^{n+1} \cdot n!}{(n+1)! \cdot x^n}\bigg| < 1$

$\displaystyle |x| \cdot \lim_{n \to \infty} \dfrac{1}{n+1} < 1$

$|x| \cdot 0 < 1$ for all $x \in \mathbb{R}$ ... interval of convergence $(-\infty,\infty)$