Maclaurin polynomial degree question

[James150]

Hi,

What does it mean to write a maclaurin polynomial of f(x) = cos(2x) to degree n = 2k? While i know how to find the polynomial for f(x) = cos(2x), how does one write a polynomial to degree 2k if k is forever changing?

 

[James150]

Hi,

What does it mean to write a maclaurin polynomial of f(x) = cos(2x) to degree n = 2k? While i know how to find the polynomial for f(x) = cos(2x), how does one write a polynomial to degree 2k if k is forever changing?

I just noticed that the maclaurin formula is often written with k and n interchangeably but this just confuses me even more

 

[HallsofIvy]

First, you should realize that "n" and "k" are arbitrary choices so any formula written using "n" can be written using "k" and mean exactly the same thing. Here, however, we are given an explicit relationship, n= 2k.

Do you know the McLaurin polynomial for cos(x) using, say, "n" as the index? Do you see what you get if you replace "x" by "2x" in that polynomial? Do you notice that, since cos(x) is an "even" function, that polynomial has only even exponents? Do you see that if n= 8 it can be written as n= 2(4). So what is "k" in that case?

 

[Dr.Peterson]

Do you see what you get if you replace "x" by "2x" in that polynomial? Do you notice that, since cos(x) is an "even" function, that polynomial has only even exponents? Do you see that if n= 8 it can be written as n= 2(4). So what is "k" in that case?

I think this was supposed to be "replace n by 2k".

 

[James150]

I think this was supposed to be "replace n by 2k".

Yes i think you are right. I ended up writing out the maclaurin series for f(x) = cos(2x) up to ((-1)^k)(((2x)^2k)/(2k)!. Basically i replaced n with 2k. I was confused because i thought it was asking me to write the polynomial to degree 2k which i didnt think was possible since it is infinite and k changes every iteration. I was over thinking it

 

[HallsofIvy]

You are confusing "MacLaurin polynomial" with "MacLauring series".

A MacLaurin polynomial is NOT "infinite"- a MacLaurin series is.

The MacLaurin series for \(\displaystyle e^x\), for example, is \(\displaystyle \sum_{i=0}^\infty \frac{1}{i!}x^i\).

The MacLaurin polynomial \(\displaystyle e^x\), to degree n, is \(\displaystyle \sum_{i=0}^n\)\(\displaystyle \frac{1}{i!} x^i\).