<s>[INST] Why $ \lim\limits_{n \to \infty} \frac{ (n+1)^k}{(n+1)^{k+1}-n^{k+1}}=\frac{1}{k+1} $? Can anyone explain the following equation?
$$\lim_{n \to \infty} \frac{ (n+1)^k}{(n+1)^{k+1}-n^{k+1}}=\frac{1}{k+1}$$
 [/INST] $$\frac{(n+1)^{k}}{(n+1)^{k+1}-n^{k+1}} = \frac{1}{n+1} \frac{1}{1-(1+1/n)^{-(k+1)}}$$
Taylor expand in the denominator in the RHS:
$$\frac{(n+1)^{k}}{(n+1)^{k+1}-n^{k+1}}  \approx \frac{1}{n+1} \frac{1}{1-(1-(k+1)/n)}=\frac{n}{n+1} \frac{1}{k+1}$$
The result follows from taking the limit as $n\to\infty$.
</s>