$(\frac{n}{n+1})^n$ converges to $0$, so why is it converge to 1 when $n=999999....$?

Question

I hope to find out if it's an issue with my calculator or something more interesting.

Answer 1

Hint: $$ \left( \frac{n}{n+1} \right)^n = \frac{1}{\frac{1}{\left( \frac{n}{n+1} \right)^n}} = \frac{1}{\left( \frac{n+1}{n} \right)^n} = \frac{1}{\left( 1 + \frac{1}{n} \right)^n} $$ Can you take it from here?

Answer 2

$$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n= e^{-1}\ne0.$$ Any evidence against this from your calculator is likely to be due to inaccurate floating point arithmetic.