Consider the following ODE:
$$\frac{d^2 x}{dt^2} + kx = f(t)$$
where $$f(t) = \frac{1}{2} +\sum_{n = 1}^{\infty}{ \frac{-4}{n \pi} \sin\left(\frac{n \pi t}{2}\right)}$$ was derived using fourier series.
I have to find a particular solution. The sum symbol confuses me on this one but I will treat it as one does normally when looking to find a particular solution.
My thoughts:
Guess the solution: $$ x(t) = A_1 t+ A_0 + \sum_{n = 1}^{\infty}{ B \sin\left(\frac{n \pi t}{2}\right) + C \cos\left(\frac{n \pi t}{2}\right)} $$
After taking the second derivative and matching coefficients I arrive to the conclusion that the coefficients of the particular solution are:
$$ A_1 = 0 $$
$$ A_0 = \frac{1}{2k} $$
$$ C = 0 $$
$$ B = - \frac{4}{(k - m (\frac{n \pi}{2})^2)n \pi} $$
Should I take into consideration the sum operator when determining the coefficient $B$,
$$ B = - \frac{ \frac{4b}{ \pi } \sum_{n=1}^{\infty}{\frac{1}{n}} }{a - \sum_{n=1}^{\infty}{( \frac{n \pi}{2})^2}} $$
where $$a = \frac{k}{m}, \space b = \frac{1}{m}$$
and insert this with the sum operator (as a second sum) into my guess?