Find m.g.f. given $E(X^r)$ function? 
"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?
