有d數學問題唔識想向大家請教3 - 後傳

證明(1+e)^2/(2e)=2+1/2!+1/4!+1/6!+…
Sol
e^x=Σ(k=0 to ∞)_x^k/k!
e^(－1)= Σ(k=0 to ∞)_(－1)^k/k!
e=Σ(k=0 to ∞)_(1)^k/k!
(1+e)^2/e
=(1+2e+e^2)/e
=2+e^(－1)+e
=2+Σ(k=0 to ∞)_(－1)^k/k!+Σ(k=0 to ∞)_(1)^k/k!
=2+Σ(k=0 to ∞)_[(－1)^k+(1)^k]/k!
=2+[(－1)^0+(1)^0]/0!+Σ(k=1to ∞)_[(－1)^k+(1)^k]/k!
=4+Σ(k=1 to ∞)_[(－1)^k+(1)^k]/k!
=4+Σ(m=1 to ∞)_[(－1)^(2m)+(1)^(2m)]/(2m)!
+Σ(m=1 to ∞)_[(－1)^(2m+1)+(1)^(2m+1)]/(2m+1)!
=4+Σ(m=1 to ∞)_(1+1)/(2m)!+Σ(m=1 to ∞)_(－1+1)/(2m+1)!
=4+Σ(m=1 to ∞)_(1+1)/(2m)!
=4+2/2!+2/4!+2/6!+….
So
(1+e)^2/(2e)=2+1/2!+1/4!+1/6!+..

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2010-10-15 17:27:15 補充：
我是新手, 想試一試, 所以就亂click了, sorry ......

2010-10-15 17:30:24 補充：
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