The discussion revolves around the infinite sum \(\sum_{n=0}^{\infty}{\frac{1}{(2n)!}}\), with participants exploring its relationship to known series, particularly the exponential function and cosine series.
There is ongoing exploration of the series, with participants offering suggestions on how to approach the problem, including writing out the series for \(e\) and \(e^{-1}\) as infinite sums. Multiple interpretations of the series and its connections to other functions are being discussed.
Participants note the potential confusion with the cosine series and the implications of introducing hyperbolic functions, indicating a careful consideration of definitions and assumptions in the problem setup.
stanli121 said: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?
Perhaps more to the pointstanli121 said: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?
HallsofIvy said: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]?