For any real number x, show that x - 0.5*x^2 < ln(1+x)?

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

I suggest to form a new function `f(x) = x - x^2/2 - ln(1+x)`  and to use derivative to check is the function is negative or positive.

Differentiating the function yields:

`f'(x) = 1 - 2x/2 - 1/(1+x) =gt f'(x) = (1 + x - x - x^2 - 1)/((1+x)^2)`

Reducing like terms in brackets yields:

`f'(x) = (- x^2)/((1+x)^2)`

`` Notice that the derivative is negative for x `in`  R.

Since the derivative is negative =>`f(x) lt 0 =gt x - x^2/2 - ln(1+x) lt 0 =gt x - x^2/2lt ln(1+x).`

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The Maclaurin series for ln(1 + x) where |x| < 1 is given by

ln(1 + x) = `x - x^2/2 + x^3/3 - x^4/4` ...

From the series for ln(1+x) it can be inferred that `x - x^2/2 < x - x^2/2 + x^3/3 - x^4/4`

=> `x - x^2/2 < ln(1 + x)`

It has to be kept in mind that the series is valid only for |x|< 1.

Using the Maclaurin expansion for ln(1 + x), the given inequality can be proved

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