Show that Euler-Mascheroni sequence is decreasing &monotonic

[timnswede]

Homework Statement

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

Homework Equations

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]

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

 

[timnswede]

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]

Integrate both sides with respect to what?

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

 

[timnswede]

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]

Integrating 1/n gets me ln(n)

If you integrate from where to where?

 

[timnswede]

If you integrate from where to where?

From the beginning of the sequence, 1, to the end, n.

 

[utkarsh009]

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