Taylor's Theorem for Sin(a+x) and Proving Convergence | Homework Solution

In summary, Taylor's theorem can be used to find expansions for sin(a+x) in powers of x. The limit of the remainder term is between -1 and 1 if x is less than one.
  • #1
Lucy Yeats
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Homework Statement



Taylor's theorem can be stated f(a+x)=f(a)+xf'(a)+(1/2!)(x^2)f''(a)+...+(1/n!)(x^n)Rn
where Rn=fn(a+y), 0≤y≤x
Use this form of Taylor's theorem to find an expansion of sin(a+x) in powers of x, and show that in this case, mod([itex]\frac{x^n Rn}{n!}[/itex])[itex]\rightarrow[/itex]0 as n[itex]\rightarrow[/itex][itex]\infty[/itex] for all x.

Homework Equations


The Attempt at a Solution



sin(a+x)=sin(a)+xcos(a)-[itex]\frac{1}{2!}[/itex]x^2sin(a)-[itex]\frac{1}{3!}[/itex]x^3cos(a)...

I don't know how to prove the next bit. Also, I don't understand why Rn=fn(a+y) rather than Rn=fn(A). Any help would be great.
 
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  • #2
I'm still stuck on this. Any help would be brilliant!
 
  • #3
Hi Lucy Yeats! :smile:

##R_n=f^{(n)}(a+y)## is part of the remainder term.
Taylor's theorem states that there is such an y so that the remainder term is equal to the sum of the remaining terms in the series.

You already have the expansion for sin(a+x), although perhaps you should find a generic formula for the terms.

Furthermore you would need to find the limit of the remainder term.
Can you think of an upper and a lower bound for Rn?
 
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  • #4
Hello again!

The question says 0≤y≤x, so maybe fn(a)≤Rn≤fn(a+x)?
 
  • #5
Lucy Yeats said:
Hello again!

The question says 0≤y≤x, so maybe fn(a)≤Rn≤fn(a+x)?

That would only be true if fn would be a monotonically increasing function.
But sines and cosines are notorious for it that they are not.

What do you think ##f^{(n)}(u)## looks like, knowing that f(u)=sin(u)?
 
  • #6
fn(u) is always between 1 and -1?
 
  • #7
How did you come to that idea?
 
  • #8
Because the gradient is always sin, cos, -sin, or -cos, which have ranges between -1 and 1.
 
  • #9
Right!

So...
 
  • #10
Can I have another hint? I really can't see the next step.
 
  • #11
You are supposed to find the limit of ##|\frac{x^n R_n}{n!}|##.
What do you know and what can you say about this limit?
 
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  • #12
So mod(Rn) is between -1 and 1, so I thought about the x^n/n! part. The limit will only be zero if x is less than one. But x could be greater than 1, so I'm confused.
 
  • #13
So suppose x>1, say x=100.
What will x^n/n! be for large(r) values of n?
 
  • #14
So if x=100, 100^n>n!
 
  • #15
Lucy Yeats said:
So if x=100, 100^n>n!

How do you know that 100^n>n! ?
Is that true for every n?
 
  • #16
Is that ony true if n<x?
 
  • #17
Let's pick a smaller value for x, say x=3.

Then x^3 = 3x3x3 > 1x2x3 = n!
x^4 = 3x3x3x3 > 1x2x3x4 = n!
What do you get for n=5, 6, 7, 10, 100?
 
  • #18
Hey Lucy!

Did you give up on this thread?
That would be a pity!
 

What is Taylor's Theorem for Sin(a+x)?

Taylor's Theorem for Sin(a+x) is a mathematical formula that allows us to approximate the value of Sin(a+x) using a polynomial function. It states that any smooth function can be approximated by a polynomial function in a specific range around a given point, in this case, Sin(a+x) around the point 0.

How do you prove convergence using Taylor's Theorem for Sin(a+x)?

To prove convergence using Taylor's Theorem for Sin(a+x), we need to show that the error term, which is the difference between the actual value of Sin(a+x) and the value obtained from the polynomial approximation, approaches 0 as the degree of the polynomial increases. This means that the polynomial function becomes a better and closer approximation of Sin(a+x) as the degree increases, leading to convergence.

What is the importance of Taylor's Theorem for Sin(a+x)?

Taylor's Theorem for Sin(a+x) is important because it allows us to approximate the value of Sin(a+x) with a high degree of accuracy. This is useful in various fields of science and engineering, such as physics, where sinusoidal functions are commonly used to model natural phenomena.

How is Taylor's Theorem for Sin(a+x) related to other mathematical concepts?

Taylor's Theorem for Sin(a+x) is closely related to other mathematical concepts, such as derivatives and integrals. In fact, the coefficients of the polynomial approximation in Taylor's Theorem are determined by the derivatives of the function at the given point. This allows us to use Taylor's Theorem to find the derivatives of a function at a given point.

Can Taylor's Theorem for Sin(a+x) be applied to other trigonometric functions?

Yes, Taylor's Theorem can be applied to other trigonometric functions, such as Cosine and Tangent. The only difference is the formula for the polynomial function used in the approximation. For example, for Cosine, the polynomial function would start with an even power term, while for Tangent, it would involve both odd and even power terms.

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