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Does the sum converge or diverge in Wolfram Alpha?

Alexander Povolotsky
Posted 1 year ago

Standard WolframAlpha doesn't conclude whether below sum converges or diverges

sum(prime(n)/prime(n+1)/n^2,n=1...infinity)

Using Maple notation for the input to Inverse Symbolic Calculator

evalf[16](sum(nextprime(p)/nextprime(nextprime(p))/p^2, p = 1 .. 1000000));

gives 1.116727203339173
POSTED BY: Alexander Povolotsky
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With Mathematica we can use:

 FullSimplify[Prime[n] > n Log[n], n \[Element] Integers && n > 0]
 (*True*)

Then:

  SumConvergence[(n Log[n])/(n^2 n Log[n + 1]), n]
  (*True*)
POSTED BY: Mariusz Iwaniuk

Thanks for the solution! I wish that WolframAlpha would incorporate/internalize this approach directly into the sum command to make it more straightforward - perhaps via employing elements of AI, such as GPT (Mr. Stephen Wolfram recent post described how ChatGPT could benefit from using WolframAlpha, but this will be the case of opposite). Cheers, Alexander R. Povolotsky
POSTED BY: Alexander Povolotsky
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