Let's consider a dirac comb $f(x) = \sum_{n=-\infty}^{\infty} \delta(x-nb)$. I know it has a period of $b$. So, using the definition of finding fourier coefficients, I can write:
$C_n = \frac{1}{b} \int_{0}^{b} f(x) dx = \frac{1}{b} $$\int_{0}^{b} \sum_{n=-\infty}^{\infty} \delta(x-nb) dx$.
Now, since $x$ ranges from $0$ to $b$, only $n = 0$ and $n=1$ in the sum will give non-zero contribution to $C_n$. So, I get,
$C_n = \frac{1}{b} \int_{0}^{b} \delta(x) dx + \frac{1}{b} \int_{0}^{b} \delta(x-b) e^{-ik_{m}x}dx = \frac{1}{b} + \frac{1}{b} = \frac{2}{b}$
But the fourier series section in this wikipedia article on Dirac Comb says that the coefficient is supposed to be $\frac{1}{b}$ as have many other online articles that I have read. Could anyone explain what I am getting wrong?