please help me to sum the series cos@+cos2@+cos3@+...

I assume you are familiar with Maclauren series expansion:

f(a) = f(0) + f'(0)a + \frac {f''(0)a^2}{2!} + \frac {f'''(0)a^3}{3!} + \frac {f^{iv}(0)a^4}{4!} + ...

Apply this to f(x) = \cos(x), then f(x) = \cos(2x), etc. and sum them. What do you get?

By the way, I would point out that this series does not converge, but rather with increasing values of 'n' cycles about 0 with an amplitude that depends on the value of x.