In[]:=
deploy
Wed 1 Sep 2021 13:09:03
Background for https://mathoverflow.net/a/402932/7655
In[]:=
F[s]=Sqrt1Product1+
2s
3
k
,{k,1,∞};​​ToRadicals@FullSimplify[PowerExpand[F[s]]]//TraditionalForm
Out[]//TraditionalForm=
Γ1-
3
-2
3
s

Γ
3
2
3
s
+1
Γ
2/3
(-1)
3
2
3
s
+1
In[]:=
G[s_]=Sum
k
3
k
+2s
,{k,1,∞}
Out[]=
-
1
3
RootSum1+2s+3#1+3
2
#1
+
3
#1
&,
PolyGamma[0,-#1]
1+#1
&
In[]:=
NIntegrate[F[s]G[s],{s,0,∞},PrecisionGoal20,WorkingPrecision20]
Out[]=
1.9921805635607759019