Tricky Infinite Sum

  • Thread starter stanli121
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  • #1
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Homework Statement



What is [tex]\sum_{n=0}^{\infty}{\frac{1}{(2n)!}}[/tex]?

Homework Equations


[tex]e^x[/tex] = [tex]\sum_{n=0}^{\infty}{\frac{x^n}{n!}}[/tex]


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?
 

Answers and Replies

  • #2
sylas
Science Advisor
1,647
7

Homework Statement



What is [tex]\sum_{n=0}^{\infty}{\frac{1}{(2n)!}}[/tex]?

Homework Equations


[tex]e^x[/tex] = [tex]\sum_{n=0}^{\infty}{\frac{x^n}{n!}}[/tex]


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
 
  • #3
lanedance
Homework Helper
3,304
2
have you treid working backwards from the answer to understand how they get there?
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,847
965

Homework Statement



What is [tex]\sum_{n=0}^{\infty}{\frac{1}{(2n)!}}[/tex]?

Homework Equations


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

What x gives [itex](-1)^nx^n= (-x)^n= 1[/itex]?


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?
 
  • #5
sylas
Science Advisor
1,647
7
Perhaps more to the point
[tex]cos(x)= \sum_{n=0}^\infty \frac{(-1)^nx^n}{(2n)!}[/tex]

What x gives [itex](-1)^nx^n= (-x)^n= 1[/itex]?

Um, actually
[tex]cos(x)= \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}[/tex]
This is not going to help. And let's not introduce the hyperbolic cos. He's got the answer, and now just needs to work back from there.

Cheers -- sylas
 

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