Maclaurin series of xsinx

This exercise shows user how to turn a function into a power series. Knowledge of taking derivatives, taking integrals, power series, and Maclaurin series are encouraged to ensure success on this exercise. Khan Academy Wiki Explore. Please Read!

Next: The Maclaurin Expansion of cos x. To find the Maclaurin series coefficients, we must evaluate. The coefficients alternate between 0, 1, and You should be able to, for the n th derivative, determine whether the n th coefficient is 0, 1, or From the first few terms that we have calculated, we can see a pattern that allows us to derive an expansion for the n th term in the series, which is. Because this limit is zero for all real values of x , the radius of convergence of the expansion is the set of all real numbers.

Maclaurin series of xsinx

Since someone asked in a comment, I thought it was worth mentioning where this comes from. First, recall the derivatives and. Continuing, this means that the third derivative of is , and the derivative of that is again. So the derivatives of repeat in a cycle of length 4. That is, something of the form. What could this possibly look like? We can use what we know about and its derivatives to figure out that there is only one possible infinite series that could work. First of all, we know that. When we plug into the above infinite series, all the terms with in them cancel out, leaving only : so must be. Now if we take the first derivative of the supposed infinite series for , we get. We know the derivative of is , and : hence, using similar reasoning as before, we must have. So far, we have. Now, the second derivative of is. If we take the second derivative of this supposed series for , we get.

Using some other techniques from calculus, we can prove that this infinite series does in fact maclaurin series of xsinx toso even though we started with the potentially bogus assumption that such a series exists, maclaurin series of xsinx, once we have found it we can prove that it is in fact a valid representation of. Determine what function evaluates to the given power series: The user is asked to match up the function that evaluates to the given power series. An Alternate Explanation The following Khan Acadmey video provides a similar derivation of the Maclaurin expansion for sin x that you may find helpful.

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In the previous two sections we discussed how to find power series representations for certain types of functions——specifically, functions related to geometric series. Here we discuss power series representations for other types of functions. In particular, we address the following questions: Which functions can be represented by power series and how do we find such representations? Then the series has the form. What should the coefficients be?

Maclaurin series of xsinx

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Search for courses, skills, and videos. Finding Taylor or Maclaurin series for a function. About About this video Transcript. It turns out that this series is exactly the same as the function itself! Created by Sal Khan. Want to join the conversation? Log in.

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February 13, at pm. The Math Less Traveled. Don't have an account? Spam prevention powered by Akismet. Now if we take the first derivative of the supposed infinite series for , we get. Start a Wiki. Next: The Maclaurin Expansion of cos x. The functions cos u and sin u can be expanded in with a Maclaurin series, and cos c and sin c are constants. When we take the th derivative, the constant term is going to end up being because it started out as and then went through successive derivative operations before the term disappeared:. Sign me up.

Next: The Maclaurin Expansion of cos x. To find the Maclaurin series coefficients, we must evaluate. The coefficients alternate between 0, 1, and

So far, we have. It may be helpful in other problems to write out a few more terms to find a useful pattern. Sine Taylor Series at 0 Derivation of the Maclaurin series expansion for sin x. My students have trouble with Taylor series each time I teach it, and there is something about sine that makes it the appropriate jumping point. So the derivatives of repeat in a cycle of length 4. Please Read! Ajmain Yamin Yamin says:. I write it here only because I think it plays against this exposition nicely. It turns out that this same process can be performed to turn almost any function into an infinite series, which is called the Taylor series for the function a MacLaurin series is a special case of a Taylor series. If we take the second derivative of this supposed series for , we get. This entry was posted in calculus , infinity , iteration and tagged cos , derivative , infinite , MacLaurin , series , sin , sum , Taylor. The following Khan Acadmey video provides a similar derivation of the Maclaurin expansion for sin x that you may find helpful.