# Is possible to simplify this expression?

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...

• 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. Jan 6, 2022 at 8:00
• PossibleZeroQ[% - π] returns True. Jan 6, 2022 at 8:05
• @user64494 So you think it is impossible to ask mma to simplify this expression?
– yode
Jan 6, 2022 at 8:06
• No, I don't think so. However, I don't see much sense in that simplification as I wrote in my above comment. Jan 6, 2022 at 8:48
• @Roman First time to meet this function. But why PossibleZeroQ[Log[a b] - Log[a] - Log[b]] is not True?
– yode
Jan 6, 2022 at 13:05

Clear[expr];

True
• I get a warning that Reduce is assuming the equation is True. It seems worth mentioning to me Jan 6, 2022 at 18:43
• The result is cached. Try ClearSystemCache[]; before rerunning Reduce[]. Jan 6, 2022 at 21:15
• @MichaelE2 Thanks! ClearSystemCache is new to me. Jan 6, 2022 at 21:19