- #1

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i need to find the closed form expressiosn for the next sums:

[tex]g(x)=\sum_{n=0}^{\infty}\frac {x^{2n}}{(2n)!}[/tex]

and [tex]f(x)=\sum_{n=0}^{\infty} \frac {x^{4n+1}}{4n+1}[/tex]

well for the second one i thought differentiating, i.e [tex]f'(x)= \sum_{n=0}^{\infty} x^{4n}=1/(1-x^4)=\frac {1}{(1+x^2)*(1-x^2)}[/tex], now i could integrate it by parts but in my course i haven't yet got to integrals, so i cannot use integration by parts, do you have any other way to compute this sum.

for the first one, it looks like the power series of e^x, i tried to recursively get to the expression but so far to no success.

any pointers would be helpful.

[tex]g(x)=\sum_{n=0}^{\infty}\frac {x^{2n}}{(2n)!}[/tex]

and [tex]f(x)=\sum_{n=0}^{\infty} \frac {x^{4n+1}}{4n+1}[/tex]

well for the second one i thought differentiating, i.e [tex]f'(x)= \sum_{n=0}^{\infty} x^{4n}=1/(1-x^4)=\frac {1}{(1+x^2)*(1-x^2)}[/tex], now i could integrate it by parts but in my course i haven't yet got to integrals, so i cannot use integration by parts, do you have any other way to compute this sum.

for the first one, it looks like the power series of e^x, i tried to recursively get to the expression but so far to no success.

any pointers would be helpful.

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