A possible solution to the infinite summation of sin(x)

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The discussion centers on the potential existence of a function P(x) defined on the interval [-2π, 2π] that equals the infinite summation of sin(πnx). The contributor suggests that this series may converge to a function like tangent, despite the partial sums oscillating without approaching a specific value as n approaches infinity, except at x=0. The behavior of the series raises questions about convergence and the nature of the resulting function. This exploration highlights the complexities of infinite summations in trigonometric contexts. Overall, the discussion invites further examination of the convergence properties of the series.
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So basically here's the deal:

I believe there exists a P(x) defined on [-2π, 2π]

such that over that interval P(x) = \sum^{\infty}_{n=0}[sin(πnx)]

Its weird but I have a feeling that this might converge to a function such as tangent
 
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See here. You can see that the formula for the partial sum just oscilllates and doesn't approach any value as you go to infinity, except for x=0.
 

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