Convergence

**iPod** · May 22nd 2012, 10:33 AM

I have attached the question.

I have expanded $(1+h)^n$ using the binomial theorem, and the question after it I assume they want me to use the very same theorem to prove the convergence.

So this is what I had done;

$(n+1)^{1/n}=1+h \rightarrow n+1=(1+h)^n=1+nh+\frac{n!}{2!(n-2)!}+...+h^n$

I'm not sure how to prove convergence from then on, but I am guessing the Sandwich Rule would need to be used here at one point.
Any hints/tips?

**emakarov** · May 22nd 2012, 01:20 PM

Let $h_n$ be such that $(n+1)^{1/n}=1+h_n$ . From what you wrote, $1+nh_n+n(n-1)/2\cdot h_n^2<1+n$ , so...

**iPod** · May 22nd 2012, 01:26 PM

I understand how you derived the above inequality, however I'm not sure where to go from there - I'm not sure how you apply it to derive any convergence.

**emakarov** · May 22nd 2012, 02:17 PM

Can you show that $h_n<f(n)$ for some decreasing function $f$ ? You don't have to solve the quadratic inequality $nh_n+n(n-1)/2\cdot h_n^2<n$ for $h_n$ ; just find some function $f$ .

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