"Let $X$ be a random variable with $E(X^r) = 1 / (1 + r)$, where $r = 1, 2, 3,\ldots,n$. Find the series representation for the m.g.f. of $X$, $M(t)$. Sum this series. Identify (name) the probability distribution of $X$?
As a hint, use the Taylor Formula."
The expectation is what is throwing me off here. So it's a summation? The summation doesn't converge, and I'm not aware of how to get the mgf without knowing the distribution or pdf/cdf.
The definition of mgf is
$$ M(t) = \mathbb{E}[e^{tX}] $$ and so $$ E[X^r] = M^{(r)}(0). $$
Notice that if we represent $M(t)$ as a McLaurin series, say $$ M(t) = \sum_{n=0}^\infty \frac{m_n}{n!} t^n $$ then $M^{(n)}(0) = \frac{m_n}{n!}$. We can now equate them, getting $m_n = \frac{n!}{1+n}$ and so $$ M(t) = \sum_{n=0}^\infty \frac{m_n}{n!} t^n = \sum_{n=0}^\infty \frac{t^n}{1+n}. $$
Can you take it from here?