Compute $1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \cdots + n \cdot \frac {1}{2^n} + \cdots $ I have tried to compute the first few terms to try to find a pattern but I got
$$\frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\frac{6}{64}$$
but I still don't see any obvious pattern(s). I also tried to look for a pattern in the question, but I cannot see any pattern (possibly because I'm overthinking it?) Please help me with this problem.

Let
$$ L = \sum_{n=1}^\infty \frac{n}{2^n} $$
Then,
$$ L = \frac{1}{2} + \sum_{n=2}^\infty \frac{n}{2^n} \\  
L = \frac{1}{2} + \sum_{n=1}^\infty \frac{n+1}{2^{n+1}} \\  
L = \frac{1}{2} + \frac{1}{2} \sum_{n=1}^\infty \frac{n}{2^n} + \frac{1}{2} \sum_{n=1}^\infty \frac{1}{2^n} \\  
L = \frac{1}{2} + \frac{L}{2}  + \frac{1}{2} \sum_{n=1}^\infty \frac{1}{2^n} \\  
\frac{L}{2} = \frac{1}{2}  + \frac{1}{2} \sum_{n=1}^\infty \frac{1}{2^n} \\  
\frac{L}{2} = \frac{1}{2} + \frac{1}{2} \\  
\boxed{L = 2}
$$
