I have a bit of trouble with how I should go about showing the inductive step for my induction problem. I know the general idea is to show that it can work for all numbers based on the base case but I'm still stuck on how to show it.
$\sum_{i=2}^n \frac{1}{i} \leq \frac{n}{2}$ for n $\geq$ 2
I know we have to plug in k+1 instead of n and show that it is also equal to k by using algebra. I'm just stuck on how to show it. I guess the i is the part that confusing me.
Hint:
Compare
$$\sum_{i=2}^{k+1}\frac{1}{i} \leq \frac{k+1}{2}$$
to $$\sum_{i=2}^k\frac{1}{i} \leq \frac{k}{2}.$$
On the left-hand sides, the difference is $\dfrac1{k+1}$, and on the right-hand sides, it is $\dfrac12$.
Can you conclude ?