sequences general term: u_n = 1 + 1/2 + 1/3 + ... + 1/n - ln(n)

[jk000jk]

http://prntscr.com/d0z3lj

\(\displaystyle u_n\, =\, 1\, +\, \dfrac{1}{2}\, +\, \dfrac{1}{3}\, +\, \dfrac{1}{4}\, +\, ...\, +\, \dfrac{1}{n}\, -\, \ln(n)\)

Can someone please explain how come the general term is 1/n - ln n should not be just 1/n hence the 1/2, 1/3, 1/4 also the limit is -\infty right?

 

[ksdhart2]

It looks to me as if the ln(n) is being subtracted at the very end. That is to say, the general term of the sequence is actually 1/n. But then after adding up n terms, they subtract ln(n). If this is really what's going on here, then, no:

\(\displaystyle \displaystyle \lim _{n\to \infty }\left(u_n\right)\ne -\infty \)

However, given that this appears to be just one line of a larger problem/proof/discussion, it's impossible for me to know for sure what's really going on. Please post the full and exact text of whatever it is you're looking at that generated this sequence.

 

[jk000jk]

However, given that this appears to be just one line of a larger problem/proof/discussion, it's impossible for me to know for sure what's really going on. Please post the full and exact text of whatever it is you're looking at that generated this sequence.

Well, it just an exercise that just gives some sequences and asks to show the convergence using Weierstrass criterion http://prntscr.com/d14t7p and I can't make sense of that ln.