Prove that $\{x_n\}_n$ is convergent.

Question

Here is an exercise:

Let $x_n=1+2+\dots+\frac1n-\ln n$. Prove that $\{x_n\}_n$ is convergent.

(I believe that this can be found in the site, however I cannot find immediately, so I post it here.)

The hints are much appreciated. I don't want complete proof.

Thanks for your help.

This was proven in the site quite some times. Do you know about $\gamma$; Euler's constant? — Pedro, Jun 20 '13 at 01:14

score 4 · Accepted Answer · answered Jun 20 '13 at 01:19

4

Let's present a different proof than any I've seen on this site, via a picture and a hint.

Look carefully at the following and remember that $\displaystyle \int_1^x \frac 1t dt = \ln x$:

enter image description here

Justify that the area in blue is the limit you're looking for, and for that matter the first $n$ blue bits corresponds to the $n$th term of the sequence. Show the limit exists and is finite (which can be done entirely with the picture too).

answered Jun 20 '13 at 01:19

davidlowryduda

91,687

1

Wow! A good job. Very appreciated! +10) – Paul Jun 20 '13 at 01:20

Prove that $\{x_n\}_n$ is convergent.

1 Answers1