Question Number 103763 by mathmax by abdo last updated on 17/Jul/20

calculate{ Σ_(n=0) ^∞ (−1)^n x^n }×{Σ_(n=o) ^∞  (x^(2n) /(n+1))}

Answered by mathmax by abdo last updated on 18/Jul/20

here we use Σ a_n ×Σ b_n =Σ c_n    c_n =Σ_(i+j =n)  a_i b_j  =Σ_(i=0) ^n  a_i b_(n−i)   =Σ_(i=0) ^n  (−1)^i  x^i   (x^(2(n−i)) /(n−i +1)) =Σ_(i=0) ^n  (((−1)^i )/(n−i+1)) x^(i+2n−2i)   =Σ_(i=0) ^n  (((−1)^i )/(n−i+1))x^(2n−i)  ⇒  Σ_(n=0) ^∞ (−1)^n  x^n  ×Σ_(n=0) ^∞  (x^(2n) /(n+1)) =Σ_(n=0) ^∞ Σ_(i=0) ^n  (((−1)^i )/(n−i+1)) x^(2n−i)