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)