Expected value of a random variable $X$ with pmf $p_X(k)=(1/2)^k, k = 1,2,...,$ The Statement of the Problem:
For the random variable $X$ with pmf $p_X(k)=(1/2)^k, k = 1,2,...,$
(a) Calculate $E[X]$ and $E[X(X-1)]$.
(b) Use part (a) to compute Var$[X]$.
Where I Am:
Ok, so I plugged this problem into WolframAlpha to see what it had to say, and it said that the expected value is $\approx 1.27$. But then, when I put just the explicit formula for $E[X]$ in there, it returned the following:
$$ \sum_{k=1}^{\infty} \frac{k}{2^k}=2$$
What gives?
EDIT: I figured out what I did wrong with W|A. It interpreted my variable as a normally-distributed r.v. instead of just a plain old natural number.
 A: Note that
\begin{align}
\sum_{k} \frac{k}{2^k} &= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} +  \cdots = 1\\
& \qquad \quad \frac{1}{4} + \frac{1}{8} + \cdots = \frac{1}{2} \\
& \qquad \qquad \;\;\; \frac{1}{8}  + \cdots  = \frac{1}{4}\\
 &\qquad \qquad \qquad \qquad\vdots
\end{align}
So $\sum_{k} \frac{k}{2^k} = 2$
A: Hints:
$$
E(X)=\sum_{k=1}^\infty k\frac{1}{2^k}=2\sum_{k=1}^\infty\frac{k}{2^{k+1}}\cdot
$$
To handle the RHS above, consider the function $f(x)=\sum_{k=1}^\infty\frac{1}{x^k}=\frac{1}{x-1}$ which is well-defined for $x>1$. Then
$$
\frac{1}{(x-1)^2}=-f'(x)=\sum_{k=1}^\infty\frac{k}{x^{k+1}}
$$
and so $E(X)=2\times\frac{1}{(2-1)^2}=2$. For $E[X(X-1)]$, use a similar trick:
$$
E[X(X-1)]=\sum_{k=1}^\infty k(k-1)\frac{1}{2^k}=4\sum_{k=1}^\infty\frac{k(k-1)}{2^{k+2}}=4\left(\sum_{k=1}^\infty\frac{k(k+1)}{2^{k+2}}-\frac{1}{2}\sum_{k=1}^\infty\frac{k}{2^k}\right)
$$
and consider $f''(x)$. Once you know $E[X(X-1)]$ and $E(X)$ then
$$
E(X^2)=E[X(X-1)]+E(X),\quad \text{Var}(X)=E(X^2)-E(X)^2.
$$
As for the discrepancy with Wolfram Alpha, you need to check whether you have interacted with WA correctly as $\sum_{k=1}^\infty\frac{k}{2^k}=2$ is correct.
A: For your question $X$ is basically a geometric random variable which denotes the number of trials needed to get the first success in an infinite sequence of Bernoulli trials with $P(success) = p = q = P(failure) = 0.5$.
Now,
$$
\begin{eqnarray}
E(X) &=& \sum_{k=1}^{\infty} kp^k; \quad \text{actually }p = 1/2\\
&=& p \sum_{k=1}^{\infty} kp^{k-1}\\
&=& p \sum_{k=1}^{\infty} \frac{d}{dp} p^k\\
&=& p \frac{d}{dp}\sum_{k=1}^{\infty} p^k\\
&=& p \frac{d}{dp} \left(\frac{p}{1-p}\right)\\
&=& \frac{p}{(1-p)^2}
\end{eqnarray}
$$ 
Now put $p = \frac{1}{2}$.
Proceed in a similar way to compute $Var(X)$ and $E[X(X-1)]$. 
