Calculus, 11 9 #9 2nd way, Power Series Representation

TL;DR

Learn how to write the power series expansion for 1 + X / (1 - X) using sigma notation.

Transcript

so let me show you another way to write the power series expansion for 1 plus X over 1 - x so just like my first video we will look at this as 1 over 1 - x and then plus and we can just put the X on the side times 1 over 1 - x just like that but then the different things that I'm going to do in this video is that I would just work with the sigma no... Read More

Key Insights

• 😑 The power series expansion for 1 + X / (1 - X) can be written using sigma notation, simplifying the expression.
• 👻 Matching the powers of X allows for combining the terms in the series.
• 🫰 Adjusting the index in the series involves subtracting 1 and starting from 1 instead of 0.

Questions & Answers

Q: How is the power series expansion for 1 + X / (1 - X) written using sigma notation?

The series can be written as Sigma when N goes from 0 to infinity: X^N + Sigma when N goes from 0 to infinity: X^(N+1).

Q: Why do we need to match the powers of X in the series?

Matching the powers of X allows us to combine the terms and write the series in a simplified form using sigma notation.

Q: How is the index adjusted in the series?

To adjust the index, we subtract 1 from the N power and start the index from 1 instead of 0.

Q: What is the significance of the two sigma notations in the series?

The two sigma notations represent the two terms in the power series, and the second term is repeated because one term was taken out from the first term.

Summary & Key Takeaways

• The power series expansion for 1 + X / (1 - X) can be written as a series using sigma notation.

• The series consists of two terms: one term with X to the N power and another term with X to the (N+1) power.

• To combine the terms, the powers of X must be matched, and the index must be adjusted by subtracting 1 and starting from 1 instead of 0.