∑=[3+(-1)^n]^n*x^n/n
=∑|an|*x^n {|an|=[3+(-1)^n]^n/n}
1/Rn=lim(n->∞)|a(n+1)/an|
=lim(n->∞)|{[3+(-1)^(n+1)]^(n+1)/(n+1)}/{[3+(-1)^n]^n/n}|
當n取奇數(shù)時,
1/Rn=lim(n->∞)|{[3+1]^(n+1)/(n+1)}/{[3-1]^n/n}|
=lim(n->∞)|{2^(2n+2)/(n+1)}/{2^n/n}|
=lim(n->∞)|{2^(n+2)*n/(n+1)}|
=lim(n->∞)|{2^(n+2)/(1+1/n)}|
=∞
∴收斂半徑為R=0,
x=0時,級數(shù)收斂于0,故收斂域為0點
當n取偶數(shù)時,
1/Rn=lim(n->∞)|{[3-1]^(n+1)/(n+1)}/{[3+1]^n/n}|
=lim(n->∞)|{2^(n+1)/(n+1)}/{2^(2n)/n}|
=lim(n->∞)|{2n/[2^n*(n+1)]}|
=lim(n->∞)|{2/[2^n*(1+1/n)}|
=0
∴收斂半徑為R=∞,級數(shù)發(fā)散