It's been a long time since I've done this sort of thing, so can't remember how to solve this or the specific key terms to look it up and check for duplicate answers.
$$\prod_{k=1}^{N} \left(1 - \frac{1}{k+1}\right) = \frac{1}{1+N}$$
What's this sort of thing called and what method is used to prove it?
This is called a telescoping product. All terms completely cancel except for the first and last.
$$ \prod_{k=1}^{N}\left( 1 - \frac{1}{k+1}\right)=\prod_{k=1}^{N}\left( \frac{k}{k+1}\right) $$
$$ =\frac1{ {2}}\cdot\frac{ {2}}{ {3}}\cdot\frac{ {3}}{ {4}}\cdot\frac{ {4}}{ {5}}\cdots\frac { {k}}{ {k+1}}\cdots\frac{{ {N-1}}}{ {N}}\frac { {N}}{N+1}$$
$$\require{cancel} =\frac1{\cancel{2}}\cdot\frac{\cancel{2}}{\cancel{3}}\cdot\frac{\cancel{3}}{\cancel{4}}\cdot\frac{\cancel{4}}{\cancel{5}}\cdots\frac {\cancel{k}}{\cancel{k+1}}\cdots\frac{{\cancel{N-1}}}{\cancel{N}}\frac {\cancel{N}}{N+1}$$
$$= \frac{1}{N+1}=\frac{1}{1+N}$$