1. ## Convergence

I have attached the question.

I have expanded $\displaystyle (1+h)^n$ 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;

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

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?

2. ## Re: Convergence

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

3. ## Re: Convergence

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.

4. ## Re: Convergence

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