# Tricky Infinite Sum

## Homework Statement

What is $$\sum_{n=0}^{\infty}{\frac{1}{(2n)!}}$$?

## Homework Equations

$$e^x$$ = $$\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$$

## The Attempt at a Solution

I understand that the given series looks like the series for e^1 but I know that isn't the correct answer. The answer (without solution) was supplied as (1/2)(e+e^-1). I can't seem to figure out the extra e^-1 part. Help?

sylas

## Homework Statement

What is $$\sum_{n=0}^{\infty}{\frac{1}{(2n)!}}$$?

## Homework Equations

$$e^x$$ = $$\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$$

## The Attempt at a Solution

I understand that the given series looks like the series for e^1 but I know that isn't the correct answer. The answer (without solution) was supplied as (1/2)(e+e^-1). I can't seem to figure out the extra e^-1 part. Help?

Try writing out e (which equals e1) and e-1 as infinite sums, using the equation you have given.

Cheers -- sylas

lanedance
Homework Helper
have you treid working backwards from the answer to understand how they get there?

HallsofIvy
Homework Helper

## Homework Statement

What is $$\sum_{n=0}^{\infty}{\frac{1}{(2n)!}}$$?

## Homework Equations

$$e^x$$ = $$\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$$
Perhaps more to the point
$$cos(x)= \sum_{n=0}^\infty \frac{(-1)^nx^n}{(2n)!}$$

What x gives $(-1)^nx^n= (-x)^n= 1$?

## The Attempt at a Solution

I understand that the given series looks like the series for e^1 but I know that isn't the correct answer. The answer (without solution) was supplied as (1/2)(e+e^-1). I can't seem to figure out the extra e^-1 part. Help?

sylas
$$cos(x)= \sum_{n=0}^\infty \frac{(-1)^nx^n}{(2n)!}$$
What x gives $(-1)^nx^n= (-x)^n= 1$?
$$cos(x)= \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}$$