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?

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).

Re: Prove a sequence {x_n} converges

Hi,

Do you have a typo in your sequence? As written, the sequence x_{n} is the defining sequence of Euler's constant gamma. Moreover, the given sequence is not increasing, but decreasing!

Let $\displaystyle H_n=\sum_{k=1}^n{1\over k}$

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

$\displaystyle \gamma\approx.577$

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?

Re: Prove a sequence {x_n} converges

Could I use the monotone convergence theorem here?

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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