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Any one know how to graph a function defined as an infinite series? I need to graph the function $$f(x)=\sum_{n=1}^{\infty}\bigg(n^{2}\tan^{-1}(x- n^{2}) +n^{2}\tan^{-1}(x+ n^{2}) \bigg) $$ $x\in \mathbb R$.

EDIT: So, as Peter suggested, we can simplify the function using $\tan^{-1} a + \tan^{-1} b = \tan^{-1}((a + b) / (1 - ab))$ to be $$f(x)=\sum_{n=1}^{\infty} n^{2} \tan^{-1}\bigg(\frac{2x}{ 1-(x^{2}-n^{4})}\bigg)$$

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    What's with the $\pm$?2012-12-08
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    OK, I fixed that. Thanks!2012-12-08
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    Oh, OK. You might want to use some arctangent difference and sum formulas that should help.2012-12-08
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    You mean something like $\tan^{-1} a + \tan^{-1} b = \tan^{-1}((a + b) / (1 - ab))$.2012-12-08
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    Indeed. ${}{}{}{}{}{}$2012-12-08
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    OK, now what about some asymptotic behaviour of $\arctan$ for small values? Note that $${{2x} \over {{n^4} + 1 - {x^2}}} \sim {1 \over {{n^4}}}$$ for any nonzero $x$.2012-12-08
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    What's the radius of convergence? If the series is convergent you can plot it up to any precision using software. If it is a formal series then things are more complex.2012-12-08
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    @tst Because of my last comment, and since $\arctan x = x +o(x)$ for $x\to 0$, the radius of convergence should be infinite, as the OP is suggesting.2012-12-08
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    ok, then I see no problem. Choose $k$ and plot $$f_k(x)=\sum_{n=1}^{k}\bigg(n^{2}\tan^{-1}(x- n^{2}) +n^{2}\tan^{-1}(x+ n^{2}) \bigg)$$ then the error is proportional to the first term that is being omitted.2012-12-08

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