Is there a closed form expression for $$ \sum_{1 \le k \le n}ke^{-t}\left(e^{k+\frac{1}{2k^3}}-e^{k-\frac{1}{2k^3}} \right)? $$ this series emerge in computation output of this system via convolution of input $u(t)$ and impulse response (inverse laplase transform of transfer function of system) $y(t)=e^{-t}*u(t)=\int e^{-(t-s)}u(s)ds$ part b of this problem
as bellow