How Do You Calculate the Maclaurin Series for f(x) = 5(x^2)sin(5x)?

In summary, to find the Maclaurin series of the function f(x) = 5(x^2)sin(5x), we first need to substitute 5x for x in the Taylor series for sinx. Then, to find ck, the power of x in the formula should be k. Therefore, for c3, we need to set 2n+3=3 and solve for n. This gives us n=0, which corresponds to c3.
  • #1
lovelyasha
2
0

Homework Statement


Find the Maclaurin series of the function f(x) = 5(x^2)sin(5x)

Homework Equations



[tex]\sum[/tex](Cn*x^n)

The Attempt at a Solution


I'm supposed to enter in c3-c7
I already know that c4 and c6 are 0 because the derivative is something*sin(0)=0

but for the odd numbered c's I am having problems...
i know that the taylor series for sinx = [tex]\sum[/tex]((-1^n)*x^(2n+1))/(2n+1)!
so i just substituted in 5x and multiplied by 5x^2 and got
5 [tex]\sum[/tex] ((-1^n)*(5^(2n+1)*x^(2n+3))/(2n+1)!

so for c3 i got 5(-1^3)(5^7)/(7)! = -5^8/7!
but I am not getting the answer right for this. Can someone please explain what I am doing wrong.
 
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  • #2
c3 corresponds to n=0 in your series, not n=3 (because you have x^(2n+3), not x^n).
 
  • #3
Putting n=3 in your formula does not give you c3. It gives you c9. If you want to find ck then the power of x in the formula should be k. The power of x in the formula is 2n+3. So if you want to find c3 set 2n+3=3. You want n=0.
 
  • #4
Wow, thank you guys! That actually makes sense. I asked my teacher and he wasn't much help, but I guess he didn't know that I was plugging in the wrong n's. I have the right answers now. Thanks again :)
 

FAQ: How Do You Calculate the Maclaurin Series for f(x) = 5(x^2)sin(5x)?

What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is used to approximate the value of a function at a specific point by using its derivatives.

How do you find the coefficients of a Taylor series?

The coefficients of a Taylor series can be found by using the derivatives of the function at the point of expansion. The general formula for finding the coefficients is:
cn = f(n)(a) / n!, where f(n)(a) is the nth derivative of the function at the point a.

What is the difference between a Taylor series and a Maclaurin series?

A Taylor series is a more general form of series expansion that can be used for any point of expansion, whereas a Maclaurin series is a specific type of Taylor series where the point of expansion is at x=0. This means that the Maclaurin series only uses the derivatives of a function at x=0 to find the coefficients.

What is the purpose of using a Taylor series?

The purpose of using a Taylor series is to approximate the value of a complicated function at a specific point by using a simpler polynomial function. This can be useful in solving problems in physics, engineering, and other fields that involve complex mathematical functions.

How do you determine the accuracy of a Taylor series?

The accuracy of a Taylor series depends on the number of terms used in the expansion. The more terms used, the closer the approximation will be to the actual value of the function. The accuracy can also be improved by choosing a point of expansion that is closer to the desired value.

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