/sci/ - Science & Math

So I am kinda stuck with Apostol.
This is part B of excercise 10 4.7
Prove by induction that for all n>= 1 we have:
[math]\sum_{k=n+1}^{2n} \frac{1}{k} = \sum_{m=1}^{2n}\frac{(-1)^{m+1}}{m}[/math]
for Induction lets assume the equality holds when n=x, so let's see if it also holds for n=x+1. Now I have searched for a solution in the internet (Pic related), now I understand the process, and the reason why I couldn't get to the answer is because I was adding a extra term to the equation, but I don't understand why I'm wrong. the summation on the left, since it's n = j+1, shouldn't the bottom side be k = j + 2, due to this in the equation I added the term [math]-/frac{1}{x+1}[/math] but this ruins the equality.
I have checked and the equality holds when you don't change the bottom part as they do in the image (it doesn't when you change it), so does this mean that the equality is untrue? or the bottom part is supposed to remain unchanged?