Approximating the Value of 1/3√(1-x) - 1 for Small x

In mathematics, when x is a number so small that higher powers of x (like x2, x3, and beyond) can be disregarded, we often use approximation techniques. This is particularly useful in calculus and related fields, where exact values can be difficult to compute.

Taylor Series Expansion and Small x

The expression given is 1/3√(1-x) - 1. To approximate this for small values of x, we can use the Taylor series expansion for the term 1/3√(1-x). The Taylor series allows us to represent functions as infinite sums of terms calculated from the function's derivatives at a single point.

Step-by-Step Derivation

First, let's rewrite the expression:

[frac{1}{3sqrt{1-x}} - 1]

For small x, we can expand the term 1/3√(1-x) using the binomial series for (1-x)^α. In this case, α -1/2. The binomial series expansion for a general (1-x)^α is given by:

[ (1-x)^α 1 αx frac{α(α-1)}{2!}x^2 frac{α(α-1)(α-2)}{3!}x^3 cdots ]

Substituting α -1/2 into the series:

[frac{1}{sqrt{1-x}}  (1-x)^{-1/2}  1 - frac{1}{2}x - frac{1}{8}x^2 - frac{1}{16}x^3   cdots]

Now, dividing the entire series by 3:

[frac{1}{3sqrt{1-x}}  frac{1}{3} left(1 - frac{1}{2}x - frac{1}{8}x^2 - frac{1}{16}x^3   cdots right)]

Simplifying further, we get:

[frac{1}{3sqrt{1-x}}  frac{1}{3} - frac{1}{6}x - frac{1}{24}x^2 - frac{1}{48}x^3   cdots]

Next, we subtract 1 from the series:

[frac{1}{3sqrt{1-x}} - 1  left(frac{1}{3} - frac{1}{6}x - frac{1}{24}x^2 - frac{1}{48}x^3   cdots right) - 1][frac{1}{3sqrt{1-x}} - 1  -frac{2}{3} - frac{1}{6}x - frac{1}{24}x^2 - frac{1}{48}x^3   cdots]

Since we are only interested in the first order term for small x, we can neglect higher-order terms (like x2, x3, etc.). Thus:

[frac{1}{3sqrt{1-x}} - 1 approx -frac{2}{3} - frac{1}{6}x]

By simplifying, we get the final approximation:

[frac{1}{3sqrt{1-x}} - 1 approx -frac{2}{3} - frac{1}{6}x]

This approximation is valid for small values of x. In practice, this can be a useful tool in various mathematical and engineering applications where quick, approximate calculations are needed.

Conclusion

In summary, the value of 1/3√(1-x) - 1 for small x can be approximated to -2/3 - 1/6x. This approximation simplifies calculations in scenarios where exact values are not necessary, such as in quick estimations or initial computer simulations.