一道高数选择题,答案好像错了,确认下,请写出解题步骤,
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一道高数选择题,答案好像错了,确认下,请写出解题步骤,
设幂级数∑(n=1 → ∞) An x^n与∑(n=1 → ∞) Bn x^n的收敛半径分别为(5^(1/2))/3与1/3,则幂级数∑(n=1 → ∞) (Bn^2/An^2) x^n的收敛半径为 ( )
A.5
B.(5^(1/2))/3
C.1/3
D.1/5
我认为是选D
设幂级数∑(n=1 → ∞) An x^n与∑(n=1 → ∞) Bn x^n的收敛半径分别为(5^(1/2))/3与1/3,则幂级数∑(n=1 → ∞) (Bn^2/An^2) x^n的收敛半径为 ( )
A.5
B.(5^(1/2))/3
C.1/3
D.1/5
我认为是选D
![一道高数选择题,答案好像错了,确认下,请写出解题步骤,](/uploads/image/z/931345-25-5.jpg?t=%E4%B8%80%E9%81%93%E9%AB%98%E6%95%B0%E9%80%89%E6%8B%A9%E9%A2%98%2C%E7%AD%94%E6%A1%88%E5%A5%BD%E5%83%8F%E9%94%99%E4%BA%86%2C%E7%A1%AE%E8%AE%A4%E4%B8%8B%2C%E8%AF%B7%E5%86%99%E5%87%BA%E8%A7%A3%E9%A2%98%E6%AD%A5%E9%AA%A4%2C)
设:
lim(n→∞) A(n+1)/An =ρ1 ==> R1= 1/ρ1=(5^(1/2))/3
lim(n→∞) B(n+1)/Bn =ρ2 ==> R2= 1/ρ2=1/3
则:
lim(n→∞) [(B(n+1)^2/A(n+1)^2)]/(Bn^2/An^2)
=lim(n→∞) [(B(n+1)^2/Bn^2)]/(A(n+1)^2/An^2)
=ρ2^2/ρ1^2
=R1^2/R2^2
=((5^(1/2))/3)^2/(1/3)^2
=ρ
∑(n=1 → ∞) (Bn^2/An^2) x^n的收敛半径
R= 1/ρ =1/5
选 D
lim(n→∞) A(n+1)/An =ρ1 ==> R1= 1/ρ1=(5^(1/2))/3
lim(n→∞) B(n+1)/Bn =ρ2 ==> R2= 1/ρ2=1/3
则:
lim(n→∞) [(B(n+1)^2/A(n+1)^2)]/(Bn^2/An^2)
=lim(n→∞) [(B(n+1)^2/Bn^2)]/(A(n+1)^2/An^2)
=ρ2^2/ρ1^2
=R1^2/R2^2
=((5^(1/2))/3)^2/(1/3)^2
=ρ
∑(n=1 → ∞) (Bn^2/An^2) x^n的收敛半径
R= 1/ρ =1/5
选 D