Using known Maclaurin series to approximate modification of original

In summary, to find the Maclaurin polynomial P5(x) for f(x)=xsin(x/2), you can use the Maclaurin series for sin(x) and substitute x/2 for x. This will result in x\sum\frac{(-1)^{k}x^{2k+1}}{(2k + 1)!}. You can improve this further by sweeping the x inside the sum and including the summation bounds, which are n=0 to infinity.
  • #1
phosgene
146
1

Homework Statement



Recall that the Maclaurin series for sin(x) is [itex]\sum\frac{(-1)^{k}x^{2k+1}}{(2k + 1)!}[/itex].

Use this formula to find the Maclaurin polynomial P5(x) for f(x)=xsin(x/2).

Homework Equations


The Attempt at a Solution



I know that to approximate sin(x/2) with the Maclaurin polynomial for sinx, I just substitute x/2 for x. But for xsinx, since the Maclaurin series is approximating sinx, can I just substitute the series for sinx so that I get [itex]x\sum\frac{(-1)^{k}x^{2k+1}}{(2k + 1)!}[/itex]?
 
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  • #2


yes you can :smile:
 
  • #3


Yes. You can make that better if you now sweep the x inside the sum. You may also want to include the summation bounds as you can sometimes simplify further by shifting them.
 
  • #4


Sorry, didn't see that I forgot the bounds. It's supposed to be from n=0 to infinity. Thanks for the help guys :)
 

FAQ: Using known Maclaurin series to approximate modification of original

What is a Maclaurin series and how is it used to approximate modifications of an original function?

A Maclaurin series is a mathematical representation of a function using a series of terms. It is used to approximate modifications of an original function by using known coefficients and powers of the variable. This allows for a more efficient and accurate way to calculate the values of a modified function without having to use the original function directly.

What is the difference between a Maclaurin series approximation and the actual function value?

A Maclaurin series approximation is an estimation of the actual function value using a limited number of terms. The more terms that are included in the series, the closer the approximation will be to the actual value. However, the actual function value will always be more accurate than the approximation, especially for functions with more complex modifications.

Can a Maclaurin series be used for any type of function modification?

No, a Maclaurin series can only be used for modifications that can be represented as a power series. This means that the function must be able to be written in the form of a polynomial with a finite number of terms.

How do you determine the number of terms needed for an accurate Maclaurin series approximation?

The number of terms needed for an accurate Maclaurin series approximation depends on the desired level of accuracy and the complexity of the modification. Generally, the more terms that are included, the more accurate the approximation will be. However, using too many terms can also lead to computational errors and inefficiency.

Are there any limitations to using a Maclaurin series to approximate modifications of an original function?

Yes, there are some limitations to using a Maclaurin series. One limitation is that it can only be used for modifications that can be represented as a power series. Additionally, as the modifications become more complex, the number of terms needed for an accurate approximation may increase significantly, making it impractical to use. It is important to carefully consider the limitations and potential errors when using a Maclaurin series for approximation.

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