Find expectation for $x$

Question

for $f(x) = \frac{1}{(x+1)(x+2)}$, $x = 0,1,2,3,\dots$ Find $E(X)$

So far I have got to Summation infinity $x = 0$ to $x = \inf\{\frac{x}{(x+1)(x+2)}\}$ How do you work out the summation? The answer they gave is infinity which doesnt make sense.

Answer 1

I assume by $f(x)$ you mean the probability mass function of random variable $X$. The definition of the expectation in this discrete setting is $\mathbb{E}(X)=\sum \limits_{x=0}^\infty xf(x) = \sum \limits_{x=0}^\infty \frac{x}{(x+1)(x+2)}$.

This partial fraction may be split as $\sum \limits_{x=0}^\infty (\frac{2}{x+2} - \frac{1}{x+1})$. The first and second terms sum up the same basic fractions starting at different limits and scales. Noting that $\sum \limits_{x=0}^\infty \frac{2}{x+2}= \sum \limits_{x=1}^\infty \frac{2}{x+1} = -2+\sum \limits_{x=0}^\infty \frac{2}{x+1}$, our overall expectation sum simplifies to $\mathbb{E}(X) = -2 + \sum_{x=0} \limits^\infty (\frac{2}{x+1} -\frac{1}{x+1})$.

This is nothing but $-2 + \sum \limits_{x=0}^\infty \frac{1}{x+1}$, and this series is well known to diverge, specifically to $+ \infty$. To see why, visit https://en.wikipedia.org/wiki/Harmonic_series_(mathematics).