第1个回答 2015-09-02
收敛半径 R = lim<n→∞>(n+1)3^(n+1)/(n3^n) = 3
收敛域 x∈[-3,3)
S(x) = ∑<n=1,∞>x^n/(n3^n), S(0) = 0
S'(x) = ∑<n=1,∞>x^(n-1)/3^n
= ∑<n=1,∞>(1/3)(x/3)^(n-1)
= (1/3)/(1-x/3), x∈[-3,3)
S(x) = ∫<0, x> S'(t)dt +S(0)
= ∫<0, x> (1/3)dt/(1-t/3)
= -ln(1-x/3), x∈[-3,3)
∑<n=1,∞>(-1)^(n+1)/(n3^n)
= -∑<n=1,∞>(-1)^n/(n3^n)
= -S(-1) = ln(1+1/3) = 2ln2-ln3本回答被提问者和网友采纳
收敛域 x∈[-3,3)
S(x) = ∑<n=1,∞>x^n/(n3^n), S(0) = 0
S'(x) = ∑<n=1,∞>x^(n-1)/3^n
= ∑<n=1,∞>(1/3)(x/3)^(n-1)
= (1/3)/(1-x/3), x∈[-3,3)
S(x) = ∫<0, x> S'(t)dt +S(0)
= ∫<0, x> (1/3)dt/(1-t/3)
= -ln(1-x/3), x∈[-3,3)
∑<n=1,∞>(-1)^(n+1)/(n3^n)
= -∑<n=1,∞>(-1)^n/(n3^n)
= -S(-1) = ln(1+1/3) = 2ln2-ln3本回答被提问者和网友采纳
相似回答