The forum discussion centers on proving the equation $$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$ and transforming it to $$1-\frac{1}{k+2}$$ using mathematical induction. The user struggles with the algebraic manipulation, particularly in simplifying the expression $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$. A key insight provided is the identification of a sign error in the user's calculations, suggesting that expanding the expression $$1-\frac{(k+2)+1}{(k+1)(k+2)}$$ will clarify the mistake.
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You have a sign error. If you can't find it, try expandingmikky05v said:Homework Statement
I have been working on this proof for a few hours and I can not make it work out.
$$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$
i need to get to $$1-\frac{1}{k+2}$$
I get as far as $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$
then I have tried $$1-\frac{(k+2)+1}{(k+1)(k+2)}$$ bu multiplying the left fraction by (k+2) which got me nowhere.
What am I doing wrong?