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I want to find the maximum to a Poisson distribution using calculus. How can I do this? I want to do this algebraically.

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    When you say "maximum", do you mean the mode? – 2017-01-09

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Hint: The ratio of two consecutive Poisson probabilities is $\frac{p_{n+1}}{p_n}=\frac{e^{-\mu}\mu^{n+1}/(n+1)!}{e^{-\mu}\mu^{n}/n!}=\frac{\mu}{n+1}$. For what $n$ is this ratio greater/less than one?

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The Poisson distribution is only defined for $k\in\mathbb{N}$ and therefore can't be differentiated.

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    but one can graph it as a continuous function, i.e. $y=\frac{m^x e^{-m}}{x!}$. how could I find the maxima of this graph for positive $m$? – 2017-01-09
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One way to show the maximum is using the following method: $\sum_{x=0}^{\infty}e^{−µ}\frac{µ^x}{x!}$ = $e^{−µ}\sum_{x=0}^{\infty}\frac{µ^x}{x!}$ = $e^µe^{-µ}$ = 1

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    As far as I can tell, all you've shown is that the distribution is properly normalized. I see no statement as to why this is helpful for finding the maximum. – 2017-01-09