What is the limit of difference between harmonic series and natural logarithm of n+1? I'm an underg

Nubydayclellaumvcd

Nubydayclellaumvcd

Answered question

2022-05-13

What is the limit of difference between harmonic series and natural logarithm of n+1?
I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series ( A n = 1 + 1 2 + . . . + 1 n ) and this natural logarithm L = ln ( 1 + n ) and asks me to see what happens with the difference A n L when n . I've already done some symbolic computations in Matlab and the answer I get is "eulergamma". So, after some online research I found out that eulergamma is the Euler-Mascheroni constant (being 0.5772....). Not knowing too much about this, I answered the question as follows: "I notice that lim n ( A n L ) behave same as lim n ( A n ln ( n ) ) which is equal to Euler-Mascheroni constant. So, the requested difference A n L is equal to Euler-Mascheroni constant as n ."
Teacher asked me "Why does this A n L behave in the same way as this A n ln ( n )? Try to do something. Add/subtract ln ( n ) for example."
Afterall here I am and asking for your help. I tried to add/subtract ln ( n ) but nothing came up. Could you please help me with this, or at least give me some clues/hints? I don't have such an experience on this!!
Thank you for your attention!

Answer & Explanation

allstylekvsvi

allstylekvsvi

Beginner2022-05-14Added 16 answers

We have
A n L = k = 1 n 1 k log ( n + 1 ) = k = 1 n 1 k log ( n + 1 ) + log n = log ( 1 + 1 n ) log n = k = 1 n 1 k log n log ( 1 + 1 n )
Now since lim n log ( 1 + 1 n ) = 0 then
lim n k = 1 n 1 k log n = lim n A n L = γ

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