How Accurate Is the Binomial Approximation for Small x Values?

[Bucky]

Show that if x is small then

1/(1+x) - root(1-2x) ~= (3/2)x^2


im not sure how to even begin this question. there was a part 1 but i don't think its relevant. Small numbers just confuse me...how small is small in any case?

 

[HallsofIvy]

You titled this "binomial Approximations"- obviously you are intended to expand both $\frac{1}{1+x}= (1+x)^{-1}$ and $\sqrt{1- 2x}= (1-2x)^{\frac{1}{2}}$ using the binomial formula. Then drop higher powers since if x is small, x to a power is much smaller.
"how small is small in any case?" Well, in this case, small enough that the third power is negligible- because you were asked to show that this is approximately a number times x²

 

[Architeuthis Dux]

I can provide some insights on how to approach this question. Firstly, it is important to understand the concept of binomial approximation. This approximation method is used to approximate the value of a polynomial expression when the variable is small.

In this case, the expression is (1/(1+x) - root(1-2x)) and the variable is x. To show that the approximation (3/2)x^2 is valid for small values of x, we can use mathematical techniques such as Taylor series expansion or the binomial theorem.

Using Taylor series expansion, we can write the expression as:

1/(1+x) = 1 - x + x^2 + ...

and

root(1-2x) = 1 - x + (3/2)x^2 + ...

Substituting these values in the original expression, we get:

1/(1+x) - root(1-2x) = (1 - x + x^2 + ...) - (1 - x + (3/2)x^2 + ...)
= (1 - 1) + (x - x) + (x^2 - (3/2)x^2) + ...
= (3/2)x^2 + ...

Hence, we can see that as x becomes smaller and smaller, the higher order terms in the series become negligible and the expression becomes approximately equal to (3/2)x^2.

Similarly, using the binomial theorem, we can expand the expression as:

1/(1+x) - root(1-2x) = (1 + x + x^2 + ...) - (1 - x + (3/2)x^2 + ...)
= (1 - 1) + (x - x) + (x^2 - (3/2)x^2) + ...
= (3/2)x^2 + ...

Therefore, we can conclude that for small values of x, the expression (1/(1+x) - root(1-2x)) can be approximated as (3/2)x^2. The exact value of x that can be considered as "small" may vary depending on the context and the level of precision required. However, generally, values of x less than 0.1 or even 0.01 can be considered as small in this case.

In conclusion, binomial approximation is a useful tool in mathematics and can be used to approximate