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The sum of a logarithmic harmonic series

Posted 9 years ago

If I type this into wolframAlpha

sum log(k)/(k)  for k = 2..n

then I get StieltjesGamma(1)-?_1(1+n) where it also tells me that ?_1 is the StieltjesGamma(1) function. (It's strange that it did not simplify, but OK.)

However, when I go to verify the equation at n=100, it does not work out.

StieltjesGamma(1)-StieltjesGamma(1)*(1+100)
7.2815845483676724860586375874901319137736338334337952

but

sum log(k)/k  for k = 2..100
10.553976183549153710596114240658388851398614872416869

Is this a bug? It does not make much sense to me. Any ideas what is going on?

POSTED BY: Alex Townsend

Hi,

Sum[Log[k]/k, {k, 2, N}]
(*StieltjesGamma[1] - StieltjesGamma[1, 1 + N]*)

Note that the second time the functions called with two arguments. If you evaluate:

StieltjesGamma[1] - StieltjesGamma[1, 101] // N
(*10.554*)

which is the same as

Sum[Log[k]/k, {k, 2, 100}]

You are typing in

StieltjesGamma[1] - StieltjesGamma[1]*(1 + 100) // N

instead with evaluates to 7.28158.

So this is not a bug. You just evaluated different function. Note that StieltjesGamma is overloaded: you can use it with one or two input slots.

Cheers,

Marco

POSTED BY: Marco Thiel
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