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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 enter image description here

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    What is $e^{-t}$ doing there? Can't we just factor it out?2012-12-30
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    @CalvinLin Yes.2012-12-30
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    It's very unlikely that such an expression will have a closed form. Why do you ask?2012-12-30
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    The definition of the convolution transform is $$(f*g)(t)\overset{\text{def}}{=}\int_{-\infty}^{\infty}f(s)g(t-s)ds=\int_{- \infty}^{\infty}f(t-s)g(s)ds.$$ As I do not understand convolutions very well, I do not know if your integral requires the lowerbound and upperbound present in the definition. As such, I didn't include that in the edit. Is this of any relevance?2012-12-31
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    Thanks very much Dear Limitless!2013-01-01
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    I am not enough familiar with latex! so i hope you know what i say!2013-01-01

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