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I was doing an assignment for my sophomore year course of Measure and Integration, and I encountered this problem.

Show that following series is convergent $\forall a\in \mathbb{R}$ $$\sum_{k=1}^{\infty}\frac{a^k}{k!(k+3)}$$ hence show that if the function $f: \mathbb{R}\to \mathbb{R}$ is a measurable function, then following function is so $$g(x)=\sum_{k=1}^{\infty}\frac{\left(f(x)\right)^k}{k!(k+3)}$$

I have no idea how to prove the second part. In the first part, I have shown that the series is bounded by $e^a$. How do I conclude convergence from that? And please help with the second part.

Ashutosh Purohit
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Maybe you can use that pointwise limits of measurable functions are measurable. Can you show that the partial sums are measurable functions?

Adrian Portillo Fernández
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  • Sorry, I have no idea about the second part :( – Ashutosh Purohit May 05 '20 at 16:36
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    Try using the ideas in these posts https://math.stackexchange.com/questions/541118/proving-that-sum-of-two-measurable-functions-is-measurable\ https://math.stackexchange.com/questions/1169122/composition-of-measurable-continuous-functions-is-it-measurable. In the second one, your continuous function will be $x^k$ – Adrian Portillo Fernández May 05 '20 at 16:45