Exact value of infinity sum of Fourier series coefficients

[Inertigratus]

Homework Statement

Sorry for doing another thread but I can't edit the old one any longer and I found out I made some calculation error but I'm pretty sure it's right now.

The problem is to find the exact value of the series.

Homework Equations

$\sum a_{k}$
The summation is to be done from minus infinity to infinity.

$a_{k} =$-$\frac{12}{49}$cos(nπ) - $\frac{16}{7}$cos(n$\frac{π}{14}$) + $\frac{26}{49}$cos(n$\frac{π}{2}$)

The Attempt at a Solution

I was googling some and read that this sum results from when t = 0, x(0).
Then the Fourier series changes to $\frac{a_{0}}{2}$ + $\sum a_{k}$.
However, isn't the sum in the Fourier series from n = 1 to infinity? While here it's from minus to plus infinity. Can I change the index?
I was thinking that this sum is the same, as the sum from minus infinity to 1 plus the sum from 1 to infinity. And since the coefficients only have cosines which are even functions then the sum from minus infinity to 1 is equal to the sum from 1 to infinity. Therefor the whole sum is simply equal to 2 times the sum from 1 to infinity. Is this correct?
If it is, then I think I can solve it.

 

[kurt.physics]

EDIT:I just did the calculation and got the answer as 0.However, I'm not sure if it's correct and I would like to get some feedback.