Let where

Prove that the sequence converges.

Hint: Show, first, that for any . Then show that the sequence increases and . Use the theorem about the convergence and divergence of p-series to complete the proof.

How should I proceed?

Printable View

- October 1st 2013, 06:08 PMvidomagruProve a sequence {x_n} converges
Let where

Prove that the sequence converges.

Hint: Show, first, that for any . Then show that the sequence increases and . Use the theorem about the convergence and divergence of p-series to complete the proof.

How should I proceed? - October 2nd 2013, 01:48 AMchiroRe: 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). - October 2nd 2013, 06:53 AMjohngRe: 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

- October 2nd 2013, 01:08 PMvidomagruRe: Prove a sequence {x_n} converges
- October 2nd 2013, 05:16 PMvidomagruRe: Prove a sequence {x_n} converges
Could I use the monotone convergence theorem here?

- October 3rd 2013, 06:49 AMjohngRe: 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