期考提 數列

[metastable (亞穩)]

The Binomial Theorem implies that

                          (2n)!       n
(1-x)^-(1/2) = 1 +Σ   ------------- X
                  n=1   4^n *(n!)^2
(b)Estimate the error if one uses x = -1/4
and the first five non-zero terms in (a) to approximate 1/√5
<sol>
(b)                                     (2n)!        n
1/√5 =(1/2)((1-(-1/4))^-1/2= (1/2)Σ ------------- X
                                    n=0  4^n *(n!)^2
Since it is an alternating series (2 points)
                    (2n)!        n         10!
∣1/√5 - (1/2)Σ ------------- X ∣≦ ---------------
               n=0  4^n *(n!)^2        2 *4^10 (5!)^2

那個1/2是怎麼出來的 還有下面那段我也看不太懂 可以解釋一下嗎

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◆ From: 140.114.123.88
※ 編輯: metastable      來自: 140.114.123.88       (06/24 09:42)
推 suhorng:1/√[1-(-1/4)] = 1/√(4/5) = 2/√5
→ suhorng:下面那段就alternating series誤差估計
→ suhorng:誤差不超過略去的第一項
推 newversion:樓上 1/√(5/4) 才對
→ suhorng:感謝樓上