Determine the converge or divergence of the sequence

[merced]

$$a_n$$ = $$(1 + k/n)^n$$

Determine the converge or divergence of the sequence. If it is convergent, find its limit.

My professor said to convert the sequence to f(x) and use ln (ln y) and L'Hospital's Rule.

Do I have to use ln? Is there another way to find the convergence?

 

[courtrigrad]

$$e \equiv \lim_{x\rightarrow \infty}(1+\frac{1}{x})^{x}$$

 

[merced]

Oh, I didn't see that.
Thanks

 

[quasar987]

You could show that $b_n=(1 + 1/n)^n$ converges by showing it is bounded and increasing. This implies that for any k, $(b_n)^k$ converges, right? Then, with $c_n=kn$, notice that your a_n is such that

$$a_{c_n}=(b_{n})^k$$

That is to say, there is a subsequence of a_n that converges. This shows that a_n converges (towards the limit of $(b_n)^k$) because a_n is increasing, so it suffices to show that some subsequence converges to prove that the whole sequence does too.

But this is all probably more complicated then you prof's advice, eh?

 

[merced]

Yep...it worked the way my professor suggested.