The Binomial Series

As I have said, it is difficult to work out Taylor series for functions if we do not have an easy way to work out all the derivatives of the function. That is why I did not try to work out the Maclaurin series for tan x--just you try to work out the 20^th derivative of tan x.

One more class of functions that we can cope with are those of the form f (x) = (1 + x)^a. If we work out the first few derivatives we get

f'(x) = a(1 + x)^{a - 1},        f''(x) = a(a - 1)(1 + x)^{a - 2},

f'''(x) = a(a - 1)(a - 2)(1 + x)^{a - 3}

The pattern is obvious and there is no problem in writing down the general derivative.

Notice that if a is a positive whole number the derivatives are eventually all zero. This is not true if a is any other number (apart from a = 0).

We now have the Maclaurin series

$$\boxed{(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + \frac{a(a-1)(a-2)(a-3)}{4!}x^4 + \cdots}$$

If a is not a non-negative integer then this expansion is only valid for -1 < x < 1.

This expansion is called the Binomial Series.

Putting a = - 1 we get the special case

$${\frac{1}{1+x}}$$ = 1 - x + x² - x³ + x⁴ - x⁵ + ^...

which is just the usual geometric series.

Putting a = ${\frac{1}{2}}$ we get

$$\sqrt{1+x}$$ = 1 + $${\textstyle\frac{1}{2}}$$x - $${\textstyle\frac{1}{8}}$$x² + $${\textstyle\frac{1}{16}}$$x³ - $${\textstyle\frac{5}{128}}$$x⁴ + $${\textstyle\frac{7}{256}}$$x⁵ - ^...