# Prove a sequence {x_n} converges

• Oct 1st 2013, 06:08 PM
vidomagru
Prove a sequence {x_n} converges
Let $\displaystyle x_n = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} - \ln(n)$ where $\displaystyle n = 1,2,3,...$

Prove that the sequence $\displaystyle {x_n}$ converges.

Hint: Show, first, that $\displaystyle x > \ln(1+x)$ for any $\displaystyle x > 0$. Then show that the sequence $\displaystyle {x_n}$ increases and $\displaystyle x_{n+1} - x_n < \frac{1}{2(n+1)^2}$. Use the theorem about the convergence and divergence of p-series to complete the proof.

How should I proceed?
• Oct 2nd 2013, 01:48 AM
chiro
Re: Prove a sequence {x_n} converges
Hey vidomagru.

If you can show that the difference is < 1/(2(n+1))^2 then the ratio test should be sufficient to show convergence of the sequence.

Think of a power series and how the ratio test is used to show convergence (in terms of an+1/an where an is the nth coefficient).

The other hint: Use the x > ln(1+x) to bound the difference that takes into account the logarithm term (i.e. ln(1+x) - ln(x) < x+1-x = 1).
• Oct 2nd 2013, 06:53 AM
johng
Re: Prove a sequence {x_n} converges
Hi,
Do you have a typo in your sequence? As written, the sequence xn is the defining sequence of Euler's constant gamma. Moreover, the given sequence is not increasing, but decreasing!
Let $\displaystyle H_n=\sum_{k=1}^n{1\over k}$

$\displaystyle \gamma=\lim_{n\rightarrow\infty}H_n-\text{ln}(n)=\lim_{n\rightarrow\infty}x_n$

$\displaystyle \gamma\approx.577$
• Oct 2nd 2013, 01:08 PM
vidomagru
Re: Prove a sequence {x_n} converges
Quote:

Originally Posted by chiro
Hey vidomagru.

If you can show that the difference is < 1/(2(n+1))^2 then the ratio test should be sufficient to show convergence of the sequence.

Think of a power series and how the ratio test is used to show convergence (in terms of an+1/an where an is the nth coefficient).

The other hint: Use the x > ln(1+x) to bound the difference that takes into account the logarithm term (i.e. ln(1+x) - ln(x) < x+1-x = 1).

Ok, I am going to try to figure this part out and will get back to you on that, but in the mean time it seems like the sequence is decreasing not increasing?
• Oct 2nd 2013, 05:16 PM
vidomagru
Re: Prove a sequence {x_n} converges
Could I use the monotone convergence theorem here?
• Oct 3rd 2013, 06:49 AM
johng
Re: Prove a sequence {x_n} converges
Hi again,
I guess this is really the sequence you want. Yes, the monotone convergence theorem applies. Here's a complete proof:

Attachment 29371