Use power series operations to find the Taylor series at x = 0...

Question

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.

sin x – x + (x³ / 3!)

Accepted Answer

Welcome back everyone. Find the Taylor series at x equals 14 e to the power of x minus 1 minus x minus x2 divided by 2. For this problem, let's begin with the Taylor series at x equals 1. For e to the power of x, let's remember that it can be written as sigma from n equals 0 up to infinity of x to the power of n divided by n factorial, and we can basically expand it to show that it's 1 + x + x 2 divided by 2 factorial. Plus x cubed divided by 3 factorial and so on. So in this case we have e to the power of x minus 1 minus x minus x 2 divided by 2. Which can be written as sigma. From n equals 0 up to infinity of x to the power of n divided by n factorial, this is the e to the power of x term. And then we are subtracting 1 x and x squared divided by 2. Specifically, if we think about the expansion, it is 1 + x + x2 divided by 2 factorial plus x cubed divided by 3 factorial and so on. And we are basically subtracting 1, X and X 2 divided by 2. Notice that it basically cancels out one. X and X 2 divided by 2. Remember, 2 factorial is 2, right? So this is how we cancel out the terms. And what we get is x cubed divided by 3 factorial plus x to the power of 4 divided by 4 factorial and so on. In sigma notation that would be sigma from and equals. We up to infinity of x to the power of n divided by n factorial. And this is our final answer for this problem. Thank you for watching.

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.sin x – x + (x³ / 3!)

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Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)