2
$\begingroup$

After I run this code:

8/Sum[(20 m + 3) (-1)^m (4 m)!/((4 Sqrt[2])^(4 m) (m!)^4), {m, 0, Infinity}]

256/(3 (32 HypergeometricPFQ[{1/4, 1/2, 3/4}, {1, 1}, -(1/4)] - 5 HypergeometricPFQ[{5/4, 3/2, 7/4}, {2, 2}, -(1/4)]))

But as I know, the final result is $\pi$. Is possible to simplify this result to $\pi$? As I try, the FullSimplify don't work totally and FunctionExpand will just get a expression about EllipticK...

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8
  • $\begingroup$ I find the required simplification as art for art's sake. What is $\pi$? This real irrational number is in fact the sum of a certain series or the limit of a certain sequence. $\endgroup$
    – user64494
    Jan 6, 2022 at 8:00
  • 1
    $\begingroup$ PossibleZeroQ[% - π] returns True. $\endgroup$
    – Roman
    Jan 6, 2022 at 8:05
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    $\begingroup$ @user64494 So you think it is impossible to ask mma to simplify this expression? $\endgroup$
    – yode
    Jan 6, 2022 at 8:06
  • $\begingroup$ No, I don't think so. However, I don't see much sense in that simplification as I wrote in my above comment. $\endgroup$
    – user64494
    Jan 6, 2022 at 8:48
  • $\begingroup$ @Roman First time to meet this function. But why PossibleZeroQ[Log[a b] - Log[a] - Log[b]] is not True? $\endgroup$
    – yode
    Jan 6, 2022 at 13:05

1 Answer 1

2
$\begingroup$
Clear[expr];
expr=8/Sum[(20 m + 3) (-1)^m (4 m)!/((4 Sqrt[2])^(4 m) (m!)^4), {m, 0, Infinity}];
Reduce[expr == π]

True

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4
  • $\begingroup$ I get a warning that Reduce is assuming the equation is True. It seems worth mentioning to me $\endgroup$
    – Michael E2
    Jan 6, 2022 at 18:43
  • $\begingroup$ @MichaelE2 I don't know why when we run other command and then run above code,then there are no warning message. $\endgroup$
    – cvgmt
    Jan 6, 2022 at 21:12
  • $\begingroup$ The result is cached. Try ClearSystemCache[]; before rerunning Reduce[]. $\endgroup$
    – Michael E2
    Jan 6, 2022 at 21:15
  • $\begingroup$ @MichaelE2 Thanks! ClearSystemCache is new to me. $\endgroup$
    – cvgmt
    Jan 6, 2022 at 21:19

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