Power series expansion of tan(x)

TL;DR

Learn how to find the power series expansion for tangent X by using long division with infinite polynomials.

Transcript

I'm going to show you guys how to find the power series expansion for tangent X and then remember this question was actually on my own midterm as well but I was the student and a professor you'll notice that there is no easy pattern for tangent X and the way that we have to do this is that to do long division because we know tangent X is the same a... Read More

Key Insights

• 😑 Tangent X can be expressed as sine X divided by cosine X.
• ✊ The power series expansions for sine X and cosine X can be used to find the power series expansion for tangent X.
• ✊ Long division is necessary to calculate the terms of the power series expansion for tangent X.
• 🍉 The first three nonzero terms for tangent X are X, 1/3 X^3, and 2/15 X^5.
• ✊ Tangent X has no easy pattern, so long division with infinite polynomials is used to find its power series expansion.
• ✊ The power series expansion for tangent X can include an infinite number of terms.
• ✊ The power series expansion for tangent X consists of terms with increasing powers of X.

Questions & Answers

Q: How is tangent X related to sine X and cosine X?

Tangent X is equal to sine X divided by cosine X.

Q: What is the process for finding the power series expansion for tangent X?

To find the power series expansion for tangent X, we can use the power series expansions for sine X and cosine X and perform long division with infinite polynomials.

Q: Why is long division necessary for finding the power series expansion for tangent X?

There is no easy pattern for finding the power series expansion for tangent X, so long division is needed to calculate the terms of the expansion.

Q: What are the first three nonzero terms for tangent X?

The first three nonzero terms for tangent X are X, 1/3 X^3, and 2/15 X^5.

Summary & Key Takeaways

• Tangent X can be expressed as sine X over cosine X.

• The power series expansion for sine X and cosine X can be used to find the power series expansion for tangent X.

• Long division is used to find the first three nonzero terms for tangent X.