Question

How do you find the 4-th partial sum of the infinite series `sum_(n=1)^oo(1/sqrt(n)-1/sqrt(n+1))` ?

Answer 1

`S_4=(1/sqrt{1}-1/sqrt{2})+(1/sqrt{2}-1/sqrt{3})+(1/sqrt{3}-1/sqrt{4})+(1/sqrt{4}-1/sqrt{5})`

by regrouping,

`=1+(-1/sqrt{2}+1/sqrt{2})+(-1/sqrt{3}+1/sqrt{3})+(-1/sqrt{4}+1/sqrt{4})-1/sqrt{5}`

by cancelling intermediate terms,

`=1-1/sqrt{5}`