<s>[INST] Proof by induction, $1/2 + ... + n/2^n < 2$ So I'm having trouble with proving this homework question by induction. 
$$
\frac{1}{2^1} + \frac{2}{2^2} + ... +\frac{n-1}{2^{n-1}} + \frac{n}{2^n} <2
$$
I know how to prove that the series converges to 2 (using things like the ratio method), but actually using induction is where I get confused.
Base case is easy, n=1.
$$
\frac{1}{2^1}<2
$$
Induction case we assume that
$$
\frac{1}{2^1} + \frac{2}{2^2} + ... +\frac{k-1}{2^{k-1}} + \frac{k}{2^k} <2
$$
Then we get to fun old induction. How do I show that
$$
\frac{1}{2^1} + \frac{2}{2^2} + ... +\frac{k-1}{2^{k-1}} + \frac{k}{2^k} + \frac{k+1}{2^{k+1}} <2 ?
$$
 [/INST] You can try to do the following: write 
$$ \frac{1}{2} + \frac{2}{2^2} + \cdots + \frac{k}{2^k} + \frac{k+1}{2^{k+1}} = $$
$$ \frac{1}{2} + \frac{1}{2^2} + \cdots + \frac{1}{2^k} + \frac{1}{2^{k+1}} + $$ 
$$ \frac{1}{2}\left(\frac{1}{2} + \frac{2}{2^2} + \cdots + \frac{k}{2^k}\right).$$
Use that the first summand is bounded by $1$, and, by induction hypothesis, the second one is strictly less than $\frac{1}{2} \cdot 2.$ This finishes the inductive step.
</s>