Series Expansions using the Binomial Theorem | Scoins.net | DJS

## Series Expansions using the Binomial Theorem

This page follows from Binomial Expansion 1, which is in Lower School, here..

Where that page finishes it is understood that
(x+y)
n = nC0 xn + nC1 xn-1y + nC2 xn-2y² + nC3 xn-3y³ + nC4 xn-4y+ .... + nCn-r xryn-r +.. + nCn-1 xyn-1 + yn

and    nCr = n! / r! (n-r)!

This page proceeds to replace x and y with 1 and x respectively:

(1+x) = nC0  + nC1 x + nC2 x² + nC3 x³ + nC4 x+ ....... + nCn-r xn-r +.. + nCn-1 xn-1 + xn

Expanding the notation,
(1+x)
= 1 +nx +n(n-1)x²/2! + n(n-1)(n-2)x³ /3! + n(n-1)(n-2)(n-3)x /4! +...+ n(n-1)xn-2/2! + nxn-1 +x

Check exercise for you to do: write the first five terms in each case

1    (1 + x)16

2    (1 - x)¹²

3    (1 - 2x)

4    (1 - 2x/3)

Now just accept that you can do this with negative index. Try it and see, writing the first five terms:

5    (1 + x)⁻¹

6    (1 - x) ⁻¹

7    (1 + x)⁻²

8    (1 - 2x)⁻²

This last set underpins much of the financial stuff, because when x is an interest rate, (1+x)is the compound interest. To add up a series of “interests” we often find ourselves looking at a series of the form     1 + X + X² + X³ + X   where each of these X is itself an interest calculation. It seems obvious to me that  (1 - x)⁻¹  expands to an infinite series, so the expansion is only interesting if it goes to a value, called converging, which happens when the x is smaller than one.

We say the expansion (1+x) is convergent if and only if the size of x, irrespective of sign, is so that |x| <1.

If the expansion was (a+bx)ⁿ  then it would be convergent if and only if |bx|<a, which means |x| < a/b.   Generally, once n is not a positive integer, any problem that does not start with a one is made to, so (a+bx)is changed to an (1+b/a x) , which is somehow friendlier.

This implies that the expansion also works for rational index, including negative rational index. This is true. Write four terms, up to x³:

9    (1 - 2x/3)⁻²

10   (1 + x) ⁻²/³

Quite often, the index n is a positive integer for Pure (or Core) paper C1, an integer for C2 and P2 and then rational for C3 and C4. The number sets (revise perhaps?) are n∊N, n∊Z and n∊Q.

DJS 20130214

changed font to Arial Unicode in 2018. indices may be a little smaller where not in superscript.

5   1 +(-1) x +(-1)(-2)/2! x2 +(-1)(-2)(-3)/3! x3 + (-1)(-2) (-3)(-4)/4! x4       =  1 - x + x2 - x3 + x4

6     the same but all positive   1 + x + x2 + x3 + x4    Learn this one!

7   1 +(-2) x +(-2)(-3)/2! x2 +(-2)(-3)(-4)/3! x3 + (-2) (-3)(-4)(-5)/4! x4        = 1 - 2x + 3x2 - 4x3 + 5x4

8   1 +(-2)(-2x) +(-2)(-3)/2! (-2x)2 +(-1)(-2)(-3)(-4)/3! (-2x)3 + (-1)(-2) (-3)(-4)(-5)/4! (-2x)4

= 1 + 4x + 12x2 + 32x3 + 80x4

9   1 +(-2)(-2x/3) +(-2)(-3)/2! (-2x/3)2 +(-1)(-2)(-3)(-4)/3! (-2x/3)3            = 1 +4x /3 +12x2 /9 +32x3 /27

10  1 +(-2/3) x +(-2/3)(-5/3)/2! x2 +(-2/3)(-5/3)(-8/3)/3! x3                        = 1 - 2x /3 + 5x2 /9 - 40x3 /81

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