# Binomial theorem with more than 2 terms

• I
• dyn
In summary, the conversation discusses the validity of using the binomial theorem with an infinite series as the value for x, and whether the modulus of the series needs to be less than one for convergence. The example of finding the residue of 1/z^5cos(z) about the point z=0 is also mentioned, where the binomial theorem is applied with x replaced by the infinite series and n = -1. The question is raised about whether the modulus of the power series needs to be less than one for this method to work. The conversation also touches on using limits to calculate the quotient of two infinite series. The final question is whether the convergence of the series affects the calculation of the residue using the binomial expansion.

#### dyn

Hi.
Is the binomial theorem ##(1+x)^n = 1+nx+(n(n-1)/2)x^2 + ….## valid for x replaced by an infinite series such as ##x+x^2+x^3+...## with every x in the formula replaced by the infinite series ?

If so , does the modulus of the infinite series have to be less than one for the series to converge ?

You need to make sure that all ##x^n## converge absolutely.

My question arises from an example I have just come across regarding finding the residue of ##1/(z^5cos(z))## about the point z=0. The cos(z) is written out as a power series and then it seems the binomial theorem is applied to it with x replaced by the power series and n= -1. Is the modulus of that power series less than one ?

What do you mean? ##\cos(z)## is only one series.

The example expands ##1/coz(z)## as a binomial of the form ##(1+x)^n## with ## x## represented by the infinite power series starting with ##-z^2/2!## and ##n= -1##

So the question is whether ##\dfrac{1}{\lim_{n\to \infty}\sum_{k=0}^n a_k} = \lim_{n \to \infty} \dfrac{1}{\sum_{k=0}^n a_k}##, so what do you know about ##\lim_{n \to \infty} \dfrac{f_n}{g_n}##?

You've lost me now

A power series such as ##z^5\cos z## is a limit, as every infinite series. So we have the quotient ##1## divided by that limit of partial sums. You asked whether this can be calculated by as limit of ##1## divided by those partial sums. Write down what you have, with limits instead of ##\infty##. This is only a symbol. If you want to know what you can do with it, you have to use its definition.

If I am using the binomial expansion to find the residue ie. the coefficient of the ##1/z## term does it even matter if the series converges ? Whether it converges or not I should get the correct coefficient ?