How to use telescoping series to find: $\sum_{r=1}^{n}\frac{1}{r+2}$ I am a bit confused in this one, how do i do the required modification in this case?

As many people have mentioned, this sum does not telescope. A summation series telescopes only if there is addition and subtraction involved. The addition and subtraction causes terms to cancel out. For example, the summation series $\sum_{n=0}^\infty \frac1n - \frac1{n+1}$ telescopes because the subtraction allows parts of the series to cancel with parts of the series that come after. For a product series, it should have multiplication and division so that the operations cancel each other out. For example, the product $\prod_{n=0}^\infty \frac1n*({n+1})$ uses multiplication in the previous term to cancel out with division in the current term.
