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Please have a look at the Fourier Series section, and its last equation.

Let T = 1.

After expanding the Equation

x(t) = 1 + 2cos(2∏t) + 2cos(4∏t) + 2cos(6∏t) .......

Now this does not give the original Dirac Comb.

Eg: at t = 1/2

x(1/2) = 0

But RHS

= 1 + 2cos(2∏*1/2) + 2cos(4∏*1/2) + 2cos(6∏*1/2) .......

= 1 + 2cos(∏) + 2cos(2∏) + 2cos(3∏) .......

= 1 -2 + 2 -2 + 2..........

≠ LHS

What is the problem?