(-1)^n/(n) find the sum from n=0 to n=infinity

[chupe]

Homework Statement

(-1)^n/(n)! find the sum from n=0 to n=infinity and do the same for (-1)^n/(2n)! Show all work

Homework Equations

The Attempt at a Solution

First answer is .367876. I just don't know how to expand out (2n)!. I know (n)! is expanded like: 1*1, 1*2, 1*2*3, 1*2*3*4... and etc. However I don't know how the expanded version of (2n)! works.

 

[lineintegral1]

You have likely derived Taylor Expansions for some common functions, like the exponential and trigonometric functions. Can you use these to evaluate your sum?

 

[SammyS]

It looks like (-1)ⁿ/(n)! should give 1/e = e^-1 ≈0.36787944117144233

Write the first several terms for each series. Follow lineintegral1's suggestions

Also, (2n)! = 1*2*3*4* ... *(n-2*(n-1)*(n)*(n+1)*(n+2)* ... *(2n-2)*(2n-1)*(2n).

(2*0)! = 0! = 1

(2*1)! = 2! = 2

(2*2)! = 4! = 24

(2*3)! = 6! = 720

...

 

[chupe]

Thank you so much!