Did i developed these teylor series correctly

  • Thread starter transgalactic
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In summary, the conversation discusses the derivation of polynomial expressions for sine and cosine functions, as well as calculating the remainder term for these expressions. There is also discussion about using the derivatives to build a member and defining the nth member. There is uncertainty about the correctness of the expressions for sine and cosine functions, and the calculation of the remainder term.
  • #1
transgalactic
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A.
[tex]
sin^2 x=0+x^2-\frac{x^4}{3!}+\frac{(-1)^{2n}}{{(2n+1)!}^2}{x^{4n+2}}+O(4n+2)
[/tex]
B.
[tex]
xe^x=x+x^2+\frac{x^3}{2!}+\frac{x^4}{3!}++\frac{x^{n+1}}{n!}+O(n+1)
[/tex]
C.
[tex]
xsin^3 x=0+x^4-\frac{x^6}{3!}+\frac{(-1)^{2n}}{{(2n+1)!}^2}{x^{5n+6}}+O(5n+4)
[/tex]
 
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  • #2
A:
[tex]
cos 2x=cos^2x-sin^2x=1-2sin^2x\\
[/tex]
[tex]
f(x)=sin^2x=\frac{1-cos 2x}{2}\\
[/tex]
[tex]
f(0)=0\\
[/tex]
[tex]
f'(x)=\frac{1-cos 2x}{2}=cos 2xsin2x=\frac{sin4x}{2}\\
[/tex]
[tex]
f'(0)=0\\
[/tex]
[tex]
f''(x)=2cos4x\\
[/tex]
[tex]
f^{(3)}=-8sin4x\\
[/tex]
[tex]
f^{(4)}=-32cos4x\\
[/tex]
i know to to use each derivative to build a member out of it.
how to defne the n'th member??

regarding b:
[tex]
xe^x=x+x^2+\frac{x^3}{2!}+\frac{x^4}{3!}++\frac{x^{n+1}}{n!}+O(n+1)
[/tex]
i am sure that its correct because i took the series for e^x and multiplied eah member by x.
so the only error that could be is with the remainder .
i thought that if we multiply the remainder by x we add 1 to it
where is my mistake??

regarding C:
[tex]
sin3x=3sinx-4sin^3x\\
[/tex]
[tex]
f(x)=sin^3x=\frac{3sinx-sin3x}{4}\\
[/tex]
[tex]
f(0)=0\\
[/tex]
[tex]
f'(x)=\frac{3cosx-\frac{cos3x}{3}}{4}\\
[/tex]
[tex]
f'(0)=0
[/tex]
[tex]
f''(x)=\frac{-3sinx-\frac{sin3x}{9}}{4}\\
[/tex]
[tex]
f''(0)=0
[/tex]
for every member i get 0
??

did i solved A,B correctly
where is my mistake for C

??
 

Related to Did i developed these teylor series correctly

1. How do I know if I have developed a Taylor series correctly?

To ensure that a Taylor series has been developed correctly, you can check for the following things:

  • Make sure that the function you are approximating is infinitely differentiable in the given interval.
  • Check for any pattern or relationship between the terms in the series.
  • Verify that the series converges to the original function at the given point.
  • Check if the remainder term is small enough to be negligible.
If all of these conditions are met, then it is likely that you have developed the Taylor series correctly.

2. What is the purpose of developing a Taylor series?

The purpose of developing a Taylor series is to approximate a function with a polynomial in order to simplify calculations. It allows us to represent complex functions with simpler ones that are easier to manipulate and analyze.

3. Can a Taylor series be developed for any function?

No, not every function can be approximated with a Taylor series. The function must be infinitely differentiable in the given interval for the series to converge. If the function has a singularity or a discontinuity within the interval, then a Taylor series cannot be developed.

4. How many terms should I include in a Taylor series for accurate approximation?

The number of terms needed in a Taylor series depends on the accuracy desired. The more terms included, the more accurate the approximation will be. However, including too many terms can also result in unnecessary complexity. A good rule of thumb is to include enough terms so that the remainder term is small enough to be negligible.

5. Are there any other methods for approximating functions besides Taylor series?

Yes, there are other methods for approximating functions, such as using numerical methods like interpolation or regression. However, Taylor series is a powerful tool for approximating functions, especially for analytical purposes. It allows for a more precise and systematic approach to approximation compared to other methods.

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