How to prove 1/(1*2)+1/(2*3) + … + 1/((2n-1)(2n)) = 1/(n+1) + 1/(n+2) + … + 1/(2n) by induction?

Question

Basically, the expression in a task looks like this:

I've made a thesis

Then I use an assumption, to change some part of thesis:

I replaced the red part with a right side of an assumption

Then I substracted parts, which was the same in both sides of the equation, and got this:

Finally equation

But this equation isn't true, it gives a 1 = 0 What should I change? Did I something wrong with thesis?

Answer 1

Increasing $n$ from $k$ to $k+1$ adds $\frac{1}{(2k+1)(2k+2)}$ to the right-hand side, which is $0$ if $n=0$ as it's a sum of $n$ terms. Therefore, it's $\sum_{k=1}^n\frac{1}{2k(2k-1)}$, so the original problem has a misprint (the $\frac13$ should be $\frac{1}{12}$).