# Finding closed form for the next series:

• MathematicalPhysicist
In summary, the two expressions for the next sums are both $$cosh(x)= \frac{e^x+e^{-x}}{2}[/itex]and [tex]f(x)=\sum_{n=0}^{\infty} \frac {x^{4n+1}}{4n+1}$$

#### MathematicalPhysicist

Gold Member
i need to find the closed form expressiosn for the next sums:
$$g(x)=\sum_{n=0}^{\infty}\frac {x^{2n}}{(2n)!}$$
and $$f(x)=\sum_{n=0}^{\infty} \frac {x^{4n+1}}{4n+1}$$
well for the second one i thought differentiating, i.e $$f'(x)= \sum_{n=0}^{\infty} x^{4n}=1/(1-x^4)=\frac {1}{(1+x^2)*(1-x^2)}$$, 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.

Last edited:
The first one is, in fact,
$$cosh(x)= \frac{e^x+ e^{-x}}{2}[/itex] For the second one, your idea of differentiating to get a geometric series and then integrating works nicely (after you learn to integrate, of course!) using partial fractions, not "by parts". But it gives a function involving arctan and logarithms so I doubt you will find any simple way to do that. Just in case you wanted it, the second ones [tex]\frac{1}{4} (2\arctan x -\ln (x^2-1))$$ I think.

The summand for g(x) looks similar to the summand for e^x -- this suggests you might be able to express the power series for g(x) in terms of the power series for e^x.

The summand for f(x) looks similar to the summands for log(1+x) and for arctan x -- this suggests you might be able to express the power series for g(x) in terms of the power series for log(1+x) and arctan x.

after one hour i got it by myself.
btw, iv'e got something like this:
i need to find a power series for (1+x^2)*arctg(x).
i know the power series of arctg(x), but when I am multiplying this sum with (1+x^2), i get x+sum, i.e where the sum starts from n=1, but i can't put the x into the sum. perhaps it should be this way?

What is arctg(x)?

the same as arctan(x).
it's one of those shorcuts,like for sinh, there's sh, etc.

So are you saying arctg(x) is the same as arctan(x)? Or that its a similar pattern, its the inverse of g(x)? Because that's argcosh(x).

yes arctg(x) is the same as arctan(x), now back on topic.

Well then yes, it can be the way you stated, it doesn't matter and isn't wrong.