Power series of tan^-1(x), with radius & interval of convergence, calculus 2 tutorial

TL;DR

Power series expansion and integration calculations are used to find the power series for the function in first tangent X.

Transcript

okay we're going to find the power steering suspension for the function in first tension X and remember we usually have two ways to find a power to expansion but function right the first way is we can use our best friend the second way is we can use the Taylor formula and of course whenever we can utilize our bedspread which we'll definitely do so ... Read More

Key Insights

• ✊ Power series expansion and the Taylor formula are two methods to find power series expansions.
• 😑 Algebra operations can be used to find the power series for a given expression.
• ✊ Integrating a power series can help obtain the power series for a related function.
• ✊ The convergence of the power series must be checked at the endpoints, in addition to the radius of convergence.
• ✊ The power series for the function in first tangent X can be found by integrating the power series for 1 over 1 plus X squared.
• ✊ The power series for the function in first tangent X has an interval of convergence from -1 to 1, inclusive.
• ✊ The alternating series test can be used to check the convergence of the power series.

Questions & Answers

Q: What are the two ways to find a power series expansion?

The two ways are power series expansion and the Taylor formula.

Q: How can we find the power series for 1 over 1 plus X squared?

The power series for 1 over 1 plus X squared can be obtained by performing algebra operations and manipulations on the expression.

Q: How can we find the power series for the function in first tangent X?

The power series for the function in first tangent X can be obtained by integrating the power series for 1 over 1 plus X squared.

Q: What is the condition for our power series to be legitimate?

To ensure the legitimacy of our power series, we need to include the radius of convergence and the interval convergence condition, which states that the absolute value of the input must be less than 1.

Summary & Key Takeaways

• Power series expansion and the Taylor formula are two ways to find a power series expansion.

• The power series for 1 over 1 plus X squared can be obtained by using algebra operations on the expression.

• The power series for the function in first tangent X can be obtained by integrating the power series for 1 over 1 plus X squared.