Question

[3+(-1)^n]^n*x^n/n幂级数收敛半径和收敛域

Answer 1

∑=[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}|
当n取奇数时,
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)}|
=∞
∴收敛半径为R=0,
x=0时,级数收敛于0,故收敛域为0点
当n取偶数时,
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
∴收敛半径为R=∞,级数发散