Why does $\left(1+ 1/k\right)^k$ converge to $e$ as $k \to\infty$? I came across this when learning about sequences and series. No one proved it to me tho! Is there a link anywhere? Or would it be beyond what I know? (First term of analysis)

Of course, this depends on how you define $e$. Here is something that would convince a calculus student. 
Taking logs we have:
$\log((1 + \frac{1}{k})^k) = k \log(1 + \frac{1}{k})$ 
So we can take this limit as $k\rightarrow\infty$ using l'hospitals rule. 
$$ \lim_{k\rightarrow\infty} k \log(1 + \frac{1}{k}) = \lim_{k\rightarrow\infty} \frac{\log(1 + \frac{1}{k})}{\frac{1}{k}} = \lim_{k\rightarrow\infty} \frac{-\frac{1}{k^2 + k}}{-\frac{1}{k^2}} = \lim_{k\rightarrow\infty} \frac{k^2}{k^2 + k} = 1$$
Hence, using continuity of the function $e^x$, we conclude that 
$$\lim_{k\rightarrow\infty} (1 + \frac{1}{k})^k = e^1 = e$$
