Limits of Taylor Series: Is $\sin x=x+O(x^2)$ Correct?

LagrangeEuler · 2021-05-15T10:06:58+00:00

We sometimes write that \sin x=x+O(x^3) that is correct if \lim_{x \to 0}\frac{\sin x-x}{x^3} is bounded. However is it fine that to write \sin x=x+O(x^2)?

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Thread starter LagrangeEuler
Start date May 15, 2021
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Limits Series Taylor Taylor series

In summary, a Taylor series is an infinite sum of terms used to approximate a function by evaluating its derivatives at a specific point. They are commonly used in mathematics to approximate complex functions and find the value of a function at a point without using the original function. However, the accuracy of a Taylor series is limited by its radius of convergence, which varies depending on the function being approximated. The approximation $\sin x=x+O(x^2)$ is only accurate when x is close to 0, and its accuracy can be improved by including more terms or using a different center point. Techniques such as interval arithmetic and error bounds can also be used to estimate the error in the approximation.

May 15, 2021

LagrangeEuler

We sometimes write that
[tex]\sin x=x+O(x^3)[/tex]
that is correct if
[tex]\lim_{x \to 0}\frac{\sin x-x}{x^3}[/tex]
is bounded. However is it fine that to write
[tex]\sin x=x+O(x^2)[/tex]?

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May 15, 2021

fresh_42

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2023 Award

Yes, but you give away information. You could even say ##\sin x = O(x).##

May 15, 2021

mathman

Science Advisor

Writing ##O(x^3)## says there is no ##x^2## term.

1. What is the significance of the "O(x^2)" notation in the statement "sin x=x+O(x^2)"?

The "O(x^2)" notation represents the remainder term in the Taylor series expansion of sin x. It indicates that the error between the actual value of sin x and its approximation, x, is of order x^2 or smaller.

2. How accurate is the statement "sin x=x+O(x^2)" in approximating the value of sin x?

The statement is accurate up to the second order term in the Taylor series expansion of sin x. This means that for small values of x, the approximation will be very close to the actual value of sin x. However, as x increases, the error in the approximation will also increase.

3. Can the statement "sin x=x+O(x^2)" be used for all values of x?

No, the statement is only valid for small values of x. As x approaches larger values, the error in the approximation will become more significant and the statement will no longer hold true.

4. How can the accuracy of the statement "sin x=x+O(x^2)" be improved?

The accuracy can be improved by including more terms in the Taylor series expansion, such as the third or fourth order term. This will result in a more accurate approximation for larger values of x.

5. Are there any other methods for approximating the value of sin x?

Yes, there are other methods such as using trigonometric identities or using numerical methods like the Taylor series method. However, the Taylor series method is often preferred as it provides a simple and efficient way to approximate the value of sin x.