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求级数的和∑(n=0到∞)cosnx/n!
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求级数的和∑(n=0到∞)cosnx/n!
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答案和解析
Let a = Σ(n = 0 to ∞) cos(nx)/n!and b = Σ(n = 0 to ∞) sin(nx)/n!
a + bi = Σ(n = 0 to ∞) [cos(nx) + isin(nx)]/n!
= Σ(n = 0 to ∞) [e^(inx)]/n!
= Σ(n = 0 to ∞) [e^(ix)]^n/n!
= e^[e^(ix)]
= e^(cosx + isinx)
= e^(cosx) * e^(isinx)
= e^(cosx) * [cos(sinx) + isin(sinx)]
a = Σ(n = 0 to ∞) cos(nx)/n!= Re(a + bi) = e^(cosx) * cos(sinx)
a + bi = Σ(n = 0 to ∞) [cos(nx) + isin(nx)]/n!
= Σ(n = 0 to ∞) [e^(inx)]/n!
= Σ(n = 0 to ∞) [e^(ix)]^n/n!
= e^[e^(ix)]
= e^(cosx + isinx)
= e^(cosx) * e^(isinx)
= e^(cosx) * [cos(sinx) + isin(sinx)]
a = Σ(n = 0 to ∞) cos(nx)/n!= Re(a + bi) = e^(cosx) * cos(sinx)
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