作业帮help!第24题!
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/04/20 05:32:37
help!第24题!
Sn=∑(-1)^n*x^n/(1+2^n)=∑an*x^n
1/R=lim(n->∞)|a(n+1)/an|
=lim(n->∞)|(-1)^(n+1)/[1+2^(n+1)]/(-1)^n/(1+2^n)|
=lim(n->∞)|(1+2^n)/[1+2^(n+1)]|
=lim(n->∞)|[1/2^(n+1)+2^n/2^(n+1)]/[1/2^(n+1)+1]|
=lim(n->∞)|[1/2^(n+1)+1/2]/[1/2^(n+1)+1]|
=|[0+1/2]/[0+1]|
=1/2
∴收敛半径R=2
当x=±2时,Sn=∑(-1)^n*(±2)^n/(1+2^n)=∑(±1)^n*2^n/(1+2^n)
lim(n->∞)|(±1)^(n+1)*2^(n+1)/[1+2^(n+1)]/(±1)^n*2^n/(1+2^n)|
=lim(n->∞)|2*(1+2^n)/[1+2^(n+1)]|
=lim(n->∞)|2*(1/2^n+1)/[1/2^n+2]|
=|2*(0+1)/[0+2]|
=1
∴此时,级数Sn发散
∴级数Sn的收敛域为(-2,2)
1/R=lim(n->∞)|a(n+1)/an|
=lim(n->∞)|(-1)^(n+1)/[1+2^(n+1)]/(-1)^n/(1+2^n)|
=lim(n->∞)|(1+2^n)/[1+2^(n+1)]|
=lim(n->∞)|[1/2^(n+1)+2^n/2^(n+1)]/[1/2^(n+1)+1]|
=lim(n->∞)|[1/2^(n+1)+1/2]/[1/2^(n+1)+1]|
=|[0+1/2]/[0+1]|
=1/2
∴收敛半径R=2
当x=±2时,Sn=∑(-1)^n*(±2)^n/(1+2^n)=∑(±1)^n*2^n/(1+2^n)
lim(n->∞)|(±1)^(n+1)*2^(n+1)/[1+2^(n+1)]/(±1)^n*2^n/(1+2^n)|
=lim(n->∞)|2*(1+2^n)/[1+2^(n+1)]|
=lim(n->∞)|2*(1/2^n+1)/[1/2^n+2]|
=|2*(0+1)/[0+2]|
=1
∴此时,级数Sn发散
∴级数Sn的收敛域为(-2,2)