A. Differentiate the Taylor series about 0 for the following...

Key Concepts

Function Identification from Series

Identifying a function from its power series involves recognizing the series as a standard expansion of a known function. Being familiar with common Taylor series expansions enables quick identification of functions represented by derived power series, which is a valuable skill in both pure and applied mathematics.

Interval of Convergence

The interval of convergence of a power series is the set of x-values for which the series converges to a finite value. When a power series is differentiated term by term, the new series typically retains the same radius of convergence. Understanding this concept is crucial when applying operations on power series to ensure that the manipulations are valid.

Taylor Series

A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, usually around 0 (Maclaurin series). This representation allows one to express functions in a power series form, making it easier to perform analysis such as differentiation or integration term by term.

Differentiation of Power Series

When a function is represented as a power series, it can be differentiated term by term within its interval of convergence. This property simplifies finding the derivative of the function by differentiating each power of x individually and recombining the series, ensuring that the new series represents the derivative function.

Transcript

00:01 Hi everyone, in this video, we are going to differentiate the power series for a given function, which in this case is cosine x, and we are going to find the function which the differentiated power series represents.

00:14 So we already know that given f of x equals cosine x, f prime of x will equal negative sine x.

00:27 That's a common different derivative that we all know.

00:32 So if cosine x is equal to its power series, then the derivative of cosine x should be equal to the derivative of cosine x's power series.

00:44 Right.

00:45 So once we derive this power series, the function that it represents will be the derivative of cosine x or negative sine x.

00:53 So let's go ahead and do it.

00:56 The derivative of 1 minus x squared over 2 factorial plus x to the 4.

01:04 Over 4 factorial minus x to the 6 over 6 factorial plus x to the 8 over 8 factorial and so on is 0 minus 2x over 2 factorial is just 2 plus 4x cubed over 24 minus 6x to the 5th over 720 plus 8x to the 5th over 720 plus 8x to the seventh over something horrible.

01:41 Let's see, 720 times 7 times 8 is 40 ,320.

01:49 And that just gets even more and more horrible as you add more terms.

01:54 So let's go ahead and simplify.

01:55 Those twos will cancel.

01:57 That will cancel leaving 6.

01:59 That will cancel leaving 120.

02:02 That will cancel leaving something 5 ,040.

02:07 And so on.

02:11 Okay, so the derivative of the power series 4 cosine x, which represents negative sine x, is negative x plus x, sorry, x cubed over 6, which is 3 factorial, minus x to the fifth over 120, which is 5 factorial, plus x to the 7th over 7 factorial.

02:38 Okay, sorry, i don't know.

02:40 What's happening.

02:41 There we go.

02:43 Seven factorial.

02:44 Okay.

02:45 And so on.

02:47 Well, this looks an awful lot like the power series for sine x, as it should.

02:54 It's just instead of going from positive to negative to positive to negative in the terms for the power series.

03:04 So in the power series for sine x, that would be, sorry, i don't know why my arrow's not working.

03:10 But the first term would be positive.

03:12 The second term would be negative and so on.

03:14 And in our power series here, we have the first term is negative, the second one is positive, and so on.

03:20 So to get the summation notation, which we need to find the interval of convergence, we just have to add an extra factor of negative one.

03:29 That's all.

03:30 So this is going to equal from, oh, that is an m, not an n, n equals zero to infinity of, okay, so for sine x, it's negative 1 to the n, x to the 2n plus 1.

03:54 I don't know why this is so laggy, there we go, over 2n plus 1 factorial.

04:02 So that's for sine x, but we have negative sine x.

04:04 So we just have to add an extra factor of negative 1.

04:07 So it's n plus 1.

04:10 Okay.

04:12 Now we can find the interval of convergence.

04:14 To do that, we're going to use the ratio test.

04:17 That is the limit as n goes to infinity absolute value of a sub n plus 1 over a sub n and it states that if that is less than 1 then our series converges so a sub n plus 1 will equal you know this what we're taking the summation of but instead of n we're going to have n plus 1 so it's going to be negative 1, n plus 1 plus another 1.

04:57 And i'm not going to add these things together.

05:00 I'm just going to leave it by replacing n plus 1 in for n, because it's going to make it easier to simplify later.

05:07 So replacing n with n plus 1, we get negative 1 to the n plus 1, x to the 2 times n plus 1 plus 1, divided by 2 times n plus 1, by 2 times n plus 1 plus 1, and then the factorial of that.

05:37 And then a sub n is just what is given.

05:41 So negative 1 to the n plus 1, x to the 2n plus 1 over 2n plus 1 factorial.

05:53 All right, and then we're just going to plug those into our limit.

05:55 So it's the limit as n goes to infinity of absolute value.

06:03 Negative 1, n plus 1 plus 1, x to the 2 times n plus 1 plus 1 over 2 times 2...

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