The sum of the series 11+√2+1√2+√3+1√3+√4+ up to 15 terms is:

Question

The sum of the series $\dfrac{1}{1+\sqrt2}+\dfrac{1}{\sqrt2+\sqrt3}+\dfrac{1}{\sqrt3+\sqrt4}+\cdots$ up to 15 terms is:

Accepted Answer

Rationalizing, the $n$-th term $\dfrac{1}{\sqrt n+\sqrt{n+1}}=\sqrt{n+1}-\sqrt n$.

$$S_{15}=\sum_{n=1}^{15}(\sqrt{n+1}-\sqrt n)=\sqrt{16}-\sqrt1=4-1=3.$$

Solutions · 1