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幂级数求和S=（∑π^4n/(4n+1)!)/(∑π^4n/(4n+3)!)
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幂级数求和
S=（∑π^4n/(4n+1)!)/(∑π^4n/(4n+3)!)
S=（∑π^4n/(4n+1)!)/(∑π^4n/(4n+3)!)
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答案和解析
对 k=0,1,2,3, 记 A_k = ∑π^(4n+k)/(4n+k)!
则有:
A_0 + iA_1 - A_2 - iA_3 = e^(iπ) = -1
A_0 - iA_1 - A_2 + iA_3 = e^(-iπ) = -1
两式相减得: A_1 = A_3
故 S = (A_1/π)/(A_3/π^3) = π^2
则有:
A_0 + iA_1 - A_2 - iA_3 = e^(iπ) = -1
A_0 - iA_1 - A_2 + iA_3 = e^(-iπ) = -1
两式相减得: A_1 = A_3
故 S = (A_1/π)/(A_3/π^3) = π^2
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