Let $u(x,t)$ be a solution of $u_t-u_{xx}=0$
we are given the following additional information. $$u(x,0)=\frac{e^{2x}-1}{e^{2x}+1}$$
and $u(x,t)$ is bounded.
I want to find the $\lim_{t\to \infty} u(1,t)$
I know the general solution of heat equation is given by $$u(x,t)= \sum {B_n}\sin \left( {\frac{{n\pi x}}{L}} \right){{\bf{e}}^{ - k{{\left( {\frac{{n\pi }}{L}} \right)}^2}\,t}}\hspace{0.25in}n = 1,2,3, \ldots$$
so we have $$u(x,0)=\frac{e^{2x}-1}{e^{2x}+1}=\sum{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right)$$
How should I proceed after this step.
P.S. This was asked in GATE 2006 Mathematical Sciences. Key says the answer is -1/2. Since no boundry country was given(either the question is wrong or we just have to assume the standard problem \begin{align*}& \frac{{\partial u}}{{\partial t}} = k\frac{{{\partial ^2}u}}{{\partial {x^2}}}\\ & u\left( {x,0} \right) = f\left( x \right)\hspace{0.25in}u\left( {0,t} \right) = 0\hspace{0.25in}u\left( {L,t} \right) = 0\end{align*}