Question Number 100785 by 175 last updated on 28/Jun/20

Answered by john santu last updated on 28/Jun/20

consider m ≤ (( n))^(1/(3 )) ≤ m+1 m^3 ≤ n ≤m^3 +3m^2 +1 Σ_(n = 1) ^∞ (((−1)^(n+1) )/(⌊ (√n) ⌋ )) = Σ_(m=1) ^∞ Σ_(⌊ (√( n)) ⌋=m) (((−1)^(n+1) )/m) = −Σ_(m = 1) ^∞ (((−1)^m )/m) = −[Σ_(m=1) ^∞ (((−x)^m )/m) ]_(x=0) ^(x=1) = ∫_0 ^1 Σ_(k=0) ^∞ (−x)^k dx = ∫_0 ^1 (dx/(1+x)) = [ ln(x+1) ]_0 ^1 = ln(2)