- #1

╔(σ_σ)╝

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## Homework Statement

Let [tex] x_n = \sum_{k=1}^{n}\frac{1}{k}[/tex]

Show [tex] x_n [/tex] is not cauchy.

It seems like a fairly easy problem . I bet my head is just not in the right place tonight ( It's thanksgiving in Canada :D) .

## Homework Equations

## The Attempt at a Solution

Well I know it is not bounded hence it cannot be cauchy but I doubt I am supposed to use this. I guess I am supposed to "show" it by some sort of algebraic manipulation.For n > m

[tex] |x_n - x_m| = \frac{1}{m+1} + \frac{1}{m+1} +...+ \frac{1}{n} [/tex]

[tex] |x_n -x_m| > \frac{m-n}{m+1} [/tex]I am trying to show that I can find an [tex] \epsilon >0 [/tex] for all [tex] n_0[/tex] such [tex] m,n \geq n_o[/tex] then [tex]| x_n -x_m| > \epsilon [/tex]

I am kinda stuck at this point. :(

I want to find some sort of [tex] \epsilon[/tex] in terms of [tex] n_0[/tex] but I am having a hard time.

I can find [tex] \epsilon [/tex] in terms of n,m but obviously that is not useful to me.

So I am looking for a way to relate [tex] \frac{m-n}{m+1}[/tex] to some inequality involving [tex] n_0 [/tex]

Any hints would be good.