If x is sufficiently small for powers of x above the second to be neglected. prove that 1/(1+e^(x)) = (2-x)/4
Please show the steps. Thank you very much!!!
Maths Question!!! Thx~
2012-12-02 5:17 am
回答 (2)
2012-12-02 6:16 am
✔ 最佳答案
If x is sufficiently small, exp x ~ 1 + x by Maclaurin series So, 1/(1 + exp(x))
~ 1/(1 + 1 + x)
= 1/(x + 2)
Now, as powers of x above the second to be neglected
(x + 2)(2 - x) = 4 - x^2 ~ 4
Then, (2 - x)/4 = 1/(x + 2)
2012-12-02 5:55 am
In fact, this sum is fairly simple: { }內的表示刪去!!!
1/[1+e^(x)] = (2-x)/4
LHS=1/[1+e^(x)]
=[1+e^(x)]x16e/1x16e
=[16e+16e^(x)^2]/16e
={2}(2-x){(2e)}/{16}4{e}
=(2-4)/4
RHS=(2-x)/4
Since (2-x)/4=(2-x)/4
Therefore 1/(1+e^(x)) = (2-x)/4 is an identity.
2012-12-01 21:57:06 補充：
揀我!!!
1/[1+e^(x)] = (2-x)/4
LHS=1/[1+e^(x)]
=[1+e^(x)]x16e/1x16e
=[16e+16e^(x)^2]/16e
={2}(2-x){(2e)}/{16}4{e}
=(2-4)/4
RHS=(2-x)/4
Since (2-x)/4=(2-x)/4
Therefore 1/(1+e^(x)) = (2-x)/4 is an identity.
2012-12-01 21:57:06 補充：
揀我!!!
收錄日期: 2021-04-27 17:44:06
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