Maclaurin Polynomial

  • Thread starter estro
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  • #1
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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.
 

Answers and Replies

  • #2
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.
 
  • #3
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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.
 
  • #4
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.
 
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  • #5
Char. Limit
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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...
 
  • #6
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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.
 
  • #7
vela
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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.
 

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