Find the Taylor series for the functions defined as follows. Give the interval of convergence fo

step 1 And hello, we are looking at chapter 12, section 5, problem number 5. So we want to find a Taylo

Find the Taylor series for the functions defined as follows....

Key Concepts

Taylor Series Expansion

The Taylor series is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. When the expansion point is 0, it is specifically called the Maclaurin series. This concept allows us to approximate functions by polynomials, making them easier to analyze and compute.

Geometric Series

The geometric series is a series of the form 1 + u + u^2 + u^3 + …, which converges to 1/(1 - u) when the absolute value of u is less than 1. Recognizing a function in this form allows us to express it as a power series, facilitating both analysis and computation, especially after an appropriate substitution to match the geometric series format.

Interval of Convergence

The interval of convergence of a power series is the range of input values for which the series converges to the function. It is determined by the condition under which the series, particularly after substitution into a standard series form like the geometric series, satisfies its radius of convergence criteria.

Transcript

00:01 And hello, we are looking at chapter 12, section 5, problem number 5.

00:07 So we want to find a taylor series expansion for our function f of x equals 5 over 2 minus x, and also find the interval of convergence.

00:19 So what i want to start with is the elementary tailor series for 1 over 1 minus x, since that's going to match up pretty well with what we're looking for.

00:30 So i'll write that right here.

00:33 So this is our given elementary series that we know 1 over 1 minus x.

00:38 We know is 1 plus x plus x squared plus and then it keeps going.

00:45 We get to our nth term, x to the n, and still keep going.

00:52 And this would be true for negative 1 is less than x is less than 1.

00:59 So that's its interval of convergence.

01:03 So what we need to do is alter that to make it match 5 over 2 minus x.

01:08 So we're going to do this in two steps.

01:12 So my first step i'm going to replace x with x over 2.

01:29 So that would look like 1 over 1 minus x over 2.

01:38 And so i change every term or every x in my expansion.

01:43 I change to x over 2.