Calculating Maclaurin Polynomial of 3rd Order for ln(cosx)

[estro]

I have hard time to come with Maclaurin Polynomial of a given order [lets say 3] for a composite function like ln(cosx).

Will appreciate help of how to approach such a problem.

 

[carlodelmundo]

Hi estro,

Here are some questions that can get you started:

• What is the difference between MacLaurin vs. Taylor Series? (Hint: Is MacLaurin centered at any point?)
• What does it mean for a polynomial to have order 3? (Hint: Look at the exponent)
• What is the infinite series equivalent of cos (x) and ln (x)? (Hint: Look it up in a textbook)
• What does it mean when a function is a composite of two functions? (Hint: f(g(x)) is a composition of two functions. The input of f(x) is g(x).)
• Can we multiply basic infinite series (such as sin(x), cos(x)) to create a new function? (Hint: yes)
• Can we take the composition of two series and create a new series? (Hint: yes)

From this, the problem is trivial. You can arrive to the solution in no time.

 

[estro]

Thanks for your swift response!

[*]What is the difference between MacLaurin vs. Taylor Series? (Hint: Is MacLaurin centered at any point?)

MacLaurin is a Taylor series centered about x=0.

[*]What does it mean for a polynomial to have order 3? (Hint: Look at the exponent)

MacLaurin is an infinite sum, when we talk about order 3 we take only first 3 sums.

[*]What is the infinite series equivalent of cos (x) and ln (x)? (Hint: Look it up in a textbook)

I know the series for cos(t).
I know the series for ln(1+t) so I can use the substitution [t=(cosx-1)]
But i won't write it here as I have troubles with latex.

[*]What does it mean when a function is a composite of two functions? (Hint: f(g(x)) is a composition of two functions. The input of f(x) is g(x).)
[*]Can we multiply basic infinite series (such as sin(x), cos(x)) to create a new function? (Hint: yes)
[*]Can we take the composition of two series and create a new series? (Hint: yes)
From this, the problem is trivial. You can arrive to the solution in no time.

This is where I stuck, suppose p_3(x) is 3rd degree series of cos(x) and r_3(t) is 3rd degree series for ln(1+t).
It seems logical for a 3rd order series for ln(cosx) to be something like this:
r_3(p_3(x)-1) but from this equation I get very complicated expression that seems to me wrong.

 

[carlodelmundo]

This is where I stuck, suppose p_3(x) is 3rd degree series of cos(x) and r_3(t) is 3rd degree series for ln(1+t).

As you said, if you let t = cos x - 1. Then naturally you'll get r₃(p₃(x)). Note that you forgot to cancel out the -1 (ie: 1 + -1 = 0 when substituting t for cos x - 1).

It seems logical for a 3rd order series for ln(cosx) to be something like this:
r_3(p_3(x)-1) but from this equation I get very complicated expression that seems to me wrong.

True. The expression is indeed complicated, but remember: the question asks you to write the series to the third order.

Hint: Write the first 4 or 5 terms of cos (x). Then use composition to transform the polynomial values of cos(x) for ln(x).

Keep this in mind also: The question doesn't specify to find the composite of two MacLaurin Series (ie: ln(cos(x))). It just says to find the third degree polynomial that represents this function. Technically, you can just write the first 4 terms of cos(x) and natural log each term.

 

[Char. Limit]

Wouldn't it be easier just to compute the values of the derivatives of ln(cos(x)) at x=0 than do all of that rigomarole? It's just four equations...

 

[estro]

Note that you forgot to cancel out the -1 (ie: 1 + -1 = 0 when substituting t for cos x - 1).

I didn't forget, as I used t=cosx-1

True. The expression is indeed complicated, but remember: the question asks you to write the series to the third order.

Hint: Write the first 4 or 5 terms of cos (x). Then use composition to transform the polynomial values of cos(x) for ln(x).

You mean to use r_5(x) and p_1(x)

Keep this in mind also: The question doesn't specify to find the composite of two MacLaurin Series (ie: ln(cos(x))). It just says to find the third degree polynomial that represents this function. Technically, you can just write the first 4 terms of cos(x) and natural log each term.

Wouldn't it be easier just to compute the values of the derivatives of ln(cos(x)) at x=0 than do all of that rigomarole? It's just four equations...

This is not a real problem, I asked this question with educational purpose.
Still need help with this one.

 

[vela]

It's just a bunch of algebra. Just plug one series into the other and expand, keeping only terms you know will contribute to the final answer.