Dear community-friends from Mathematica
For this problem:
a much simplier solution (than based on Gamma-functions) exists.
Will Mathematica be able to find this solution too?
Please note, I offered my support regarding this matter at least five times (without success) via wolframalpha-support.
Best regards
Ralf Steiner
Dear Mariusz
That is right. But when I (standard user) ask wolframalpha the same like you I get only these results:
Input interpretation:
sum_(k=0)^infinity (C_k/2^(2 k+1))^2 (13 k+17)
Convergence tests:
The ratio test is inconclusive.
By the comparison test, the series converges.
Partial sum formula:
sum_(k=0)^n (C_k/2^(2 k+1))^2 (13 k+17) = (29 n^3 Gamma(1/2 (2 n+3))^2+149 n^2 Gamma(1/2 (2 n+3))^2+248 n Gamma(1/2 (2 n+3))^2-4 pi Gamma(n+3)^2+132 Gamma(1/2 (2 n+3))^2)/(pi Gamma(n+3)^2)
Partial sums:
More terms
Show points
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Try putting the entire problem in Mathematica notation:
Sum[(CatalanNumber[k]/2^(2 k + 1))^2 (13 k + 17), {k, 0, Infinity}]
Dear Ralf
By doing these activities below you can prevent unnatural behavior:
1.Remove the all cache and cookies from your web browser.
2.And try again ,it should work.
Dear David
That above in Mathematica notation works, but not this (similar also in Mathematica notation):
Sum[(CatalanNumber[k]/2^(2 k + 1))^2 (17 k + 19), {k, 0, Infinity}]
Input interpretation:
sum_(k=0)^infinity (C_k/2^(2 k+1))^2 (17 k+19)
Convergence tests:
The ratio test is inconclusive.
By the comparison test, the series converges.
Partial sum formula:
sum_(k=0)^n (C_k/2^(2 k+1))^2 (17 k+19) = (25 n^3 Gamma(1/2 (2 n+3))^2+127 n^2 Gamma(1/2 (2 n+3))^2+208 n Gamma(1/2 (2 n+3))^2-2 pi Gamma(n+3)^2+108 Gamma(1/2 (2 n+3))^2)/(pi Gamma(n+3)^2)