Binomial Expansion and Maclaurin series Department of Mathematics University of Leicester

Binomial expansion The binomial expansion is a series approximation of a function: Where n is a real number.

Positive and Integer exponents Let n be any positive integer. The the series will always be of finite length. Example: Let n=2.

Pascal's triangle Pascal’s triangle can be used to solve expansions of. Pascal’s triangle has the form

Pascal's triangle Where every number is the sum of the two numbers above it.

Pascal's triangle We first take 1 on its own, then proceed in a triangle below it. The numbers on the diagonals always take the value 1.

Pascal's triangle With the 2 nd line, we start on the left. The fist number has only one above it, so is again 1. The 2 nd has 2 ones above it, so added together equal 2. And the 3 rd is again 1.

Pascal's triangle This carries on line by line:

Generate Pascal’s Triangle

Pascal's triangle To solve an expansion, we take the line in the triangle where the 2 nd and 2 nd to last element is n, and multiply each number, from left to right, by and so on.

Pascal's triangle Example: for, we take the line and multiply each element by

Positive n Any positive integer n will always have a finite number of elements. Thus being a finite sequence. This is not the same for negative integers or other real numbers that aren’t positive integers. To work these we use the Binomial expansion.

Binomial expansion The series is given by, for |x|<1:

Binomial expansion Where:

Binomial expansion As any other terms after the first 3 are very small, we can discount them. Making it an approximation. Otherwise it would go on forever. The expansion only matches the true function if |x|<1, otherwise it is a non-convergent series and therefore can not exist.

Binomial expansion: example Example: Let : So that n=-2. Rearrange so that it is of the form (1+x).

Binomial expansion: example Then using the formula: Which is valid for |x|< 2/3

Binomial expansion: example This is valid for |x|< because, as the 2 nd term in the equation, here, has to have a modulus of less than 1. Therefore: And:

Series expansion of rational functions We can expand more complicated expressions, now, using the method of partial fractions where needed: Example: let: We can split this up using partial fractions, into:

Series expansion of rational functions: example Where we can see: Then using the binomial series to expand both functions:

Series expansion of rational functions: example Therefore: Given: |x|<1

Series expansion of rational functions: example And: Given:|x|<

Series expansion of rational functions: example Therefore:

Maclaurin series The Binomial expansion is used to approximate a function of the form: as a polynomial representation. But this doesn’t work for all functions, such as that of sinx or cosx. For these we use Maclaurin series.

Maclaurin series Maclaurin series is used to approximate a function as a series around the origin, to see what a function would look like in this area. As we do the proof, bare in mind the number of assumptions needed to prove this, and that all must apply for the function to satisfy the expansion.

Maclaurin series Maclaurin series is given by:

Maclaurin series: proof Take any function f(x) and represent it as a power series: Assuming that the function can be written in this way.

Maclaurin series: proof Using this, we can equate this at x=0: Therefore:

Maclaurin series: proof This is assuming that f(x) exists at x=0: For instance: does NOT exist as x=0, and therefore cannot be used as part of this series. This also assumes that both f(x) and the polynomial representation are differentiable.

Maclaurin series: proof Now differentiate both sides: Then again equating at x=0:

Maclaurin series: proof This again assumes that f’(x) exists as 0. For all successive derivatives of f(x) in this series we will assume exists at x=0. We will also assume from now on that all successive differentials and their polynomial representation are differentiable.

Maclaurin series: proof Then differentiate again: Then equating at x=0:

Maclaurin series: proof We can find all of the by doing this. And Thus achieving the formula:

Maclaurin series: example Example: Let f(x)= Using the formula, and the fact that which at x=0,

Maclaurin series: example Then: Which we can simplify to: for all x

Maclaurin series: example Example: Let f(x)= Using a table of differentiation: Differentiated: at x=0:

Maclaurin series: example Then. Using the formula: Which we can simplify to: for all x

Maclaurin series: example Example: Let f(x)= Using a table of differentiation: Differentiated: at x=0:

Maclaurin series: example Then. Using the formula: Which we can simplify to: for all x

Maclaurin series: example Example: Let f(x)= Using a table of differentiation: Differentiated: at x=0:

Maclaurin series: example Then. Using the formula: Which we can simplify to:

Maclaurin series: example Example: Let f(x)= Using a table of differentiation: Differentiated: at x=0:

Maclaurin series: example Then. Using the formula: Which we can simplify to:

Maclaurin series: Chain rule example Example: Let f(x)= Let Then we know that, using Maclaurin series, equals:

Maclaurin series: example Then, substituting back in, we get:

Maclaurin series: example Which we can then simplify to:

Using integration and differentiation to solve other functions If we know a function, we can take it’s series expansion and use it to find the expansion for it’s integral or differential.

Using integration For finding the series expansion for an integral of f(x), we use the same expansion for f(x), but we integrate the expansion. Then add the integral of f(x) at x=0 to the series.

Using integration For example: Let f(x)=cos(x). The series expansion is: The integral of f(x) equals sin(x).

Using integration If we integrate the expansion, we get:

Using integration Then we add the integral of f(x) at x=0, which is: sin(0)=0. Therefore: Which we know is the series expansion for sin(x).

Using differentiation For finding the series expansion for a differential of f(x), we use the same expansion for f(x), but we differentiate the expansion.

Using differentiation For example: Let f(x)=cos(x). The series expansion is: The differential of f(x) equals -sin(x).

Using differentiation If we differentiate the series expansion, we get:

Using differentiation Which equals: Which we know is the series expansion for –sin(x).

Conclusion Binomial expansion and Maclaurin series are used to represent functions as a polynmial series.

Conclusion Binomial expansion: Maclaurin seires: