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Given an infinite list of numbers $\{x_i\}$ is it possible and sensible to compute the first and second derivative of $\sum_{n=1}^{\infty} x_i$?

To give more context $x_i$ are real numbers, scores between -10 to 10 of a player in a game, convergence is unknown, and i would use first derivative to know when a player got to his best\worst "peak" in his career, interpertation of the second derivative in this context is not clear to me, I will think about it, suggestions are welcome...

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    This is incredibly ill-posed. What are the $x_i$? Are they variables? If so, are they independent variables? What is your notion of convergence? If not, are they numbers? If they are numbers and the sum converges, then it would have 0 for its derivatives. – 2017-01-22
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    @CameronWilliams please see edit – 2017-01-22

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If I understood correctly, $x_i $ is a time series expressing a sequence of scores obtained by a player in a game. If this is the case, you should first try to identify a function described the behaviour of these scores over time. This can be done using nonlinear regression techniques.

Once found a function that fits your data well, you can take your derivative to identify the maximum and the minimum points and to study the behaviour of the scores.