I have to find the Fourier coefficients of $ f (x) = text {exp} (x) $ for $ -1 <x <1 $ and evaluate the value of this series to $ x = $ 2.

I've calculated the coefficients to

$$ a_n = frac {(e ^ 2-1)} {e pi ^ 2 n ^ 2 + e} (-1) ^ n $$ for the cosine-terms and

$$ b_n = frac {(1-e ^ 2) pi n} {e pi ^ 2 n ^ 2 + e} (- 1) ^ n $$

for the sinus-terms.

My first question would be, if these are correct?

Assuming they are correct, I get the next set

$$ f (x) = frac {a_0} {2} + sum_ {n geq1} frac {(- 1) ^ n} {e pi ^ 2 n ^ 2 + e}

left ((e ^ 2-1) cos ( pi n x) + (1-e ^ 2) pi n sin ( pi n x) right) $$

Now, how can I calculate its value for $ x = $ 2? In this case, the series becomes a little easier, but I do not know how to calculate its value.

And finally, when I integrate the term run by term, I do not really recover the old series anymore.

I am very grateful for any kind of help!

Thank you!