sum of the series \sum_(n=1)^(\infty ) ((1)/(\root(n)(2))-(1)/(\root(n+1)(2)))

Question

how do i find the sum of this series \sum_(n=1)^(\infty ) ((1)/(\root(n)(2))-(1)/(\root(n+1)(2)))

Answer 1

S = ∑_n=1^∞[1/sqrt(2n) - 1/sqrt(2(n + 1))]. Let take partial sum: S_n = [1/sqrt(2) - 1/sqrt(4)] + [1/sqrt(4) - 1/sqrt(6)] +

+ [1/sqrt(6) - 1/sqrt(8)] + ...+ [1/sqrt(2(n - 1)) - 1/sqrt(2n)] + [1/sqrt(2n) - 1/sqrt(2(n + 1)]. We have after collecting like terms: S_n = 1/sqrt(2) - 1/sqrt(2(n+ 1)).

So, S = lim as n → ∞ of S_n = 1/sqrt(2) because as n → ∞ 1/sqrt(2(n+ 1)) → 0.

Because S = 1/2^1/2