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The following input

s1 = Sum[(2 l + 1)*t^l*LegendreP[l, Cos[θ]], {l, 0, Infinity}, Assumptions -> {t < 1, 0 < θ < π}]

returns the closed-form expression $\frac{1-t^2}{\left(t^2-2 t \cos (\theta )+1\right)^{3/2}}$.

s2 = Sum[t^l*LegendreP[l, Cos[θ]], {l, 0, Infinity}, Assumptions -> {t < 1, 0 < θ < π}]

returns $\frac{1}{\sqrt{t^2-2 t \cos (\theta )+1}}$.

But Mathematica returns the following unevaluated:

Sum[l*t^l*LegendreP[l, Cos[θ]], {l, 0, Infinity}, Assumptions -> {t < 1, 0 < θ < π}]

The answer should be (s1 - s2)/2.

Why can't Mathematica compute this series?

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    $\begingroup$ Why can't it do s1 in the form Sum[(2 l + 1)*t^l*LegendreP[l, x], {l, 0, Infinity}, Assumptions -> {t < 1, -1 < x < 1}]? Oh, the vagaries of symbolic algorithms! $\endgroup$
    – Michael E2
    Oct 10, 2017 at 11:24
  • $\begingroup$ From this article, we know $s1$ is a special case of Eq. 21 of this paper, and $s2$ a generating function for Legendre Polynomials (see Eq. 2 of this link and Eq. 1 of this link). And both $s1$ and $s2$ is correct. $\endgroup$
    – tanghe2014
    Oct 10, 2017 at 12:15

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