How to Find Power Series Representations of ln(x) and arctan(x)

TL;DR

To find the power series representation of ln(x), derive the series for 1/x and integrate it. The radius of convergence is 1, with the interval from 0 to 2. For arctan(x), use its derivative, and determine that the interval of convergence is from -1 to 1, with convergence at both endpoints.

Transcript

consider the function ln x how can we find a power series representation of l and x in order to do this we need to realize that the integral of one over x dx is the natural log of x plus c and so we need to write a power series representation of one over x so let's do that so we need to put it in this form a over 1 minus r the sum of an infinite ge... Read More

Key Insights

• ✊ The power series representation of ln(x) can be derived by finding the power series representation of 1/x and integrating it.
• ☺️ The integration of the power series representation of 1/x helps determine the constant of integration for ln(x).
• ❓ The radius of convergence for ln(x) is 1, meaning that the series will converge for values within a distance of 1 from the center of the series.
• ❓ The interval of convergence for ln(x) is from 0 to 2, including 0 but not 2.
• ✊ The power series representation of arctan(x) can be found by using the derivative of arctan(x) and integrating it.
• ❓ The interval of convergence for arctan(x) is from -1 to 1, including both endpoints.
• 🏆 The series for arctan(x) converges at both endpoints of the interval of convergence, as determined by the alternating series test.

Questions & Answers

Q: How do we find the power series representation of ln(x)?

To find the power series representation of ln(x), we start by finding the power series representation of 1/x and integrate it. By manipulating the series and solving for the constant of integration, we can obtain the desired representation.

Q: What is the radius of convergence for ln(x)?

The radius of convergence for ln(x) is 1. This means that the series will converge for values of x within a distance of 1 from the center of the series.

Q: How do we find the interval of convergence for ln(x)?

To find the interval of convergence for ln(x), we need to test the endpoints of the interval. By plugging in values such as 0 and 2 into the series, we can determine whether they converge or not. In the case of ln(x), the interval of convergence is from 0 to 2, including 0 but not 2.

Q: How do we find the power series representation of arctan(x)?

The power series representation of arctan(x) can be found by using the derivative of arctan(x) and integrating it. By manipulating the series and simplifying, we can obtain the desired representation.

Q: What is the interval of convergence for arctan(x)?

The interval of convergence for arctan(x) is from -1 to 1, including both endpoints. This means that the series converges for values of x within this interval.

Q: Can the series for arctan(x) diverge at any point within the interval of convergence?

No, the series for arctan(x) converges at both endpoints of the interval of convergence (-1 and 1). This is determined by using the alternating series test, which shows that the series satisfies the necessary conditions for convergence.

Summary & Key Takeaways

• The power series representation of ln(x) is derived by finding the power series representation of 1/x and integrating it.

• The constant of integration for ln(x) is found by plugging in a convenient value of x (such as 1) into the power series representation.

• The radius of convergence for ln(x) is 1, and the interval of convergence is from 0 to 2.

• The power series representation of arctan(x) is found using the derivative of arctan(x) and integrating it.

• The interval of convergence for arctan(x) is from -1 to 1, and the series converges at both endpoints.