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- #1

\[

G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}.

\]

Show \(G_g\) satisfies the equation

\[

(\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')

\]

where \(\delta(x - x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\)

and the condition that

\[

\int_0^{\ell}u_m(x)G_g(x, x')dx = 0.

\]

I found

\[

u_n(x) = \sqrt{\frac{2}{\ell}}\sin(k_nx).

\]

I then end up with

\begin{gather}

(\mathcal{L} + k_m^2)G_g(x, x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx') - u_m(x)u_m(x') \\

\sum_{n\neq m}^{\infty}\sin(k_nx)\sin(k_nx') = \left(\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\right) - \sin(k_mx)\sin(k_mx')

\end{gather}

What is going wrong?