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收敛半径 R = lim<<n→∞>a<n>/a<n+1>
= lim<<n→∞>(n+1)2^(n+1)/(n2^n) = 2,
x = 2 时, 级数为 ∑<n=1,∞>1/(2n), 发散;
x = -2 时, 级数为 ∑<n=1,∞>(-1)^(n-1)/(2n), 收敛。
则原级数的收敛域是 x∈[-2,2).
记 S(x) = ∑<n=1,∞>x^(n-1)/(n2^n)
得 S(0) = 0;
当 x ≠ 0 时,
S(x) = (1/x)∑<n=1,∞>x^n/(n2^n) = S1(x)/x,
[S1(x)]' = ∑<n=1,∞>x^(n-1)/(2^n)
= (1/2)∑<n=1,∞>(x/2)^(n-1)
= (1/2)/(1-x/2) = 1/(2-x), x∈[-2,2).
S1(x) = ∫<0, x> [S1(t)]'dt +S1(0) = ∫<0, x> dt/(2-t)
= ln2 - ln(2-x) = ln[2/(2-x)], x∈[-2,2).
S(x) = (1/x) ln[2/(2-x)] , x∈[-2,2).
= lim<<n→∞>(n+1)2^(n+1)/(n2^n) = 2,
x = 2 时, 级数为 ∑<n=1,∞>1/(2n), 发散;
x = -2 时, 级数为 ∑<n=1,∞>(-1)^(n-1)/(2n), 收敛。
则原级数的收敛域是 x∈[-2,2).
记 S(x) = ∑<n=1,∞>x^(n-1)/(n2^n)
得 S(0) = 0;
当 x ≠ 0 时,
S(x) = (1/x)∑<n=1,∞>x^n/(n2^n) = S1(x)/x,
[S1(x)]' = ∑<n=1,∞>x^(n-1)/(2^n)
= (1/2)∑<n=1,∞>(x/2)^(n-1)
= (1/2)/(1-x/2) = 1/(2-x), x∈[-2,2).
S1(x) = ∫<0, x> [S1(t)]'dt +S1(0) = ∫<0, x> dt/(2-t)
= ln2 - ln(2-x) = ln[2/(2-x)], x∈[-2,2).
S(x) = (1/x) ln[2/(2-x)] , x∈[-2,2).
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