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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?
 Hi, Sum[Log[k]/k, {k, 2, N}] (*StieltjesGamma - StieltjesGamma[1, 1 + N]*) Note that the second time the functions called with two arguments. If you evaluate: StieltjesGamma - StieltjesGamma[1, 101] // N (*10.554*) which is the same as Sum[Log[k]/k, {k, 2, 100}] You are typing in StieltjesGamma - StieltjesGamma*(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