I have been attempting to convert the sum, $\sum_{n=1}^\infty$$c_n\sin(n\pi x /L)$, to a Riemann integral from a = 0 to b = $\pi$/L and with n ranging from 1 to ∞, but am a bit confused by the fact that the 'n' appears in the numerator, rather than in the denominator, of the angular part.
Can anyone help me to convert this to a continuous integral of the form normally associated with the Riemann integral for $\sum_{n=1}^\infty$$(\pi /n)\sin(i\pi /n)$?
Having already made a number of attempts, I include my latest version in the hope that someone can comment.
Using the above information, $\Delta x_k$ = $\pi$/(Ln). Since $x_k$ = k$\pi$/(Ln), then:
$\sum_{n=1}^\infty$$\ c_n$sin(n$\pi$x/L) = $\sum_{n=1}^\infty$$c_n$(Ln/$\pi$)sin(n$\pi$x/L)$\pi$/(Ln)
Substituting $\Delta x_k$ = $\pi$/(Ln) and $x_k$ = k$\pi$/(Ln), we have:
$\sum_{n=1}^\infty$$\ c_n$sin(n$\pi$x/L) = $\sum_{n=1}^\infty$$c_n$(Ln/$\pi$)sin($\pi^2$k/$L^2$)$\Delta x_k$.
From $x_k$ = $\pi$k/(Ln), we have $\pi$k/L=n$x_k$, so substituting this latter relationship into the angular part, the term on the right hand side may now simply be expressed as the integral:
$\sum_{n=1}^\infty$$\ c_n$sin(n$\pi$x/L) = $\int_0^L$$c_n$(Ln/$\pi$)sin(n$\pi$$x_k$/L)$dx_k$
which now becomes:
$\sum_{n=1}^\infty$$\ c_n$sin(n$\pi$x/L) = $\int_0^L$$c_n$(Ln/$\pi$)sin(n$\pi$$x$/L)$dx$
to which the result is:
$\sum_{n=1}^\infty$$\ c_n$sin(n$\pi$x/L) = 0
for even n, and:
$\sum_{n=1}^\infty$$\ c_n$sin(n$\pi$x/L) = 2$c_n$$L^2$/$\pi^2$
for odd n.
Any additional suggestions or comments on my results will be warmly welcomed.
Thank you in advance