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.

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.