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There is an infinite series, I want to transform it into a function, with digamma functions or something else. I hope someone can provide some guidance and suggestions. $$ E(x,y)=\sum_{n=-\infty}^{\infty} (\frac{1}{\sqrt{(2n-2x)^2+y^2}}-\frac{1}{\sqrt{(2n)^2+y^2}})， \qquad 0<x<1 $$

I have plotted this function with Matplotlib by recursion, shown below (with y = 0.025). enter image description here

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    $\begingroup$ I think this is not convergent. The terms are $\mathcal O(1/n)$ asymptotically, so it shares convergence with the harmonic series. $\endgroup$
    – PinkRabbit
    Commented Jun 14 at 11:06
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    $\begingroup$ Sorry, I made a mistake with the sign, the first term should have a positive sign. $\endgroup$
    – yongyouhe
    Commented Jun 14 at 14:46
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    $\begingroup$ I have corrected the sign. $\endgroup$
    – yongyouhe
    Commented Jun 14 at 14:52
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    $\begingroup$ The graph is misleading. For $x=0$ the 2 terms cancel, so $E(0,y)=0$, the curve should run through the origin. $\endgroup$
    – R. J. Mathar
    Commented Jun 14 at 21:13
  • $\begingroup$ The range of x should be 0 < x < 1. I forgot to add the range. $\endgroup$
    – yongyouhe
    Commented Jun 16 at 10:10

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