Power series of ln(1+x), calculus 2 tutorial

TL;DR

The video explains how to find the power series expansion for Ln of 1 plus X by connecting it to the power series of 1 over 1 plus X.

Transcript

okay we're going to find a power series expansion for the function Ln of 1 plus X and just like the previous video about finding the power series for inverse tangent X we are going to ask yourself is there any connection between our best friend and Ln of 1 plus X well if we differentiate this we get 1 over 1 plus X and this looks really similar to ... Read More

Key Insights

• ☺️ The power series for Ln of 1 plus X can be found by first finding the power series for 1 over 1 plus X and then integrating it.
• ✊ The convergence of the power series should be checked separately at the endpoints to ensure accuracy.
• ☺️ The radius of convergence for the power series of 1 over 1 plus X is 1, indicating that the series will converge for X values within a distance of 1 from the center.

Questions & Answers

Q: How is the power series for Ln of 1 plus X connected to the power series for 1 over 1 plus X?

The power series for Ln of 1 plus X is derived by first finding the power series for 1 over 1 plus X, which is differentiated from Ln of 1 plus X. By integrating the power series for 1 over 1 plus X, we can obtain the power series for Ln of 1 plus X.

Q: What is the radius of convergence for the power series of 1 over 1 plus X?

The radius of convergence for the power series of 1 over 1 plus X is 1. This means that the power series will converge for X values within a distance of 1 from the center.

Q: How is the convergence of the power series for Ln of 1 plus X checked?

The convergence of the power series for Ln of 1 plus X is checked by plugging in different X values into the series. The convergence at the endpoints is also checked separately to determine if the series converges or diverges.

Q: Can the power series for Ln of 1 plus X be used to find the value of Ln of 2?

Yes, the power series for Ln of 1 plus X can be used to find the value of Ln of 2 by plugging in X=1 into the series. However, it is important to note that the radius of convergence for the series is 1, so X values outside of this range may not yield accurate results.

Summary & Key Takeaways

• The video explains the strategy of finding the power series for Ln of 1 plus X by first finding the power series for 1 over 1 plus X and then integrating it.

• The power series for 1 over 1 plus X is derived using the Sigma notation and the expansion is shown in both expanded and summation form.

• The process of integrating the power series for 1 over 1 plus X is explained, resulting in the power series for Ln of 1 plus X.

• The convergence of the power series is discussed, with the radius of convergence being 1. The convergence at the endpoints is also checked.