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 () and this natural logarithm and asks me to see what happens with the difference when . 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 behave same as which is equal to Euler-Mascheroni constant. So, the requested difference is equal to Euler-Mascheroni constant as ."
Teacher asked me "Why does this behave in the same way as this ? Try to do something. Add/subtract for example."
Afterall here I am and asking for your help. I tried to add/subtract 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!