# Show that Euler-Mascheroni sequence is decreasing &monotonic

## Homework Statement

tn=1+1/2+1/3+...+1/n - ln(n)
a.) Interpret tn - tn+1= [ln(n+1)-ln(n)] - 1/(n+1) as a difference of areas to show that tn - tn+1 > 0.

## The Attempt at a Solution

I have not started working on part b) yet, because so far I am stuck on part a). I just simplified a bit and got ln(1+1/n)>1/(n+1). Not sure how to prove that the left side is bigger than the right.

mfb
Mentor
Did you consider an integral?
There are other methods, too, depending on the equations you got for the logarithm.

Is it as easy as just integrating both sides? I get xln(1+1/x)+ln(1+x)>ln(1+x). The left side is always greater since xln(1+1/x) is always greater than zero for n>1. Is that enough to prove it?

mfb
Mentor
Integrate both sides with respect to what?

[ln(n+1)-ln(n)] looks like an integration result.

Woops those x's up there should be n's. But with dn. I can't see what integral would have resulted in [ln(n+1)-ln(n)] though. Integrating 1/n gets me ln(n), but integrating ln(n) does not get me ln(n+1).

mfb
Mentor
Integrating 1/n gets me ln(n)
If you integrate from where to where?

If you integrate from where to where?
From the beginning of the sequence, 1, to the end, n.

From the beginning of the sequence, 1, to the end, n.
What if you integrate it from n to n+1?
Also can you think of the area of rectangle formed with height 1/(n+1) and width (n+1)-n? Just compare them on the graph of f(x)=1/x