Let LIM (n →∞) na_ N exists and the series ∑ (n = 1 →∞) n (a)_ n-a_ It is proved that the series ∑ (n = 1 →∞) a is convergent_ N convergence

Let LIM (n →∞) na_ N exists and the series ∑ (n = 1 →∞) n (a)_ n-a_ It is proved that the series ∑ (n = 1 →∞) a is convergent_ N convergence

Let the sum of the first n terms of the series ∑ n (an-a (n-1)) be: σ n
Let the sum of the first n terms of the series ∑ an be: SN
Then: σ n = nan-s (n-1) - A0
S(n-1)=nan-σn-a0
The existence of LIMS (n-1) = LIM (Nan) - Lim σ n-a0
The convergence of series ∑ an