Inverse laplace transform pf infinite product

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I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?
Polish text reading '6. Wyznacz odwrotną transformatę Laplace'a funkcji:' followed by the formula G_n(s) = product from k=1 to n of (s + k)^(-1), where k, n are in N.
 

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$$g(t) =\dfrac{ e^{-nt}}{(n-1)!} (e^t-1)^{n-1}$$

I suspect it can be proven using induction. Have a go.
 

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