Understanding an expansion into a geometric series

In summary, the conversation is about the authors describing a method for expanding 1/(a-z) using a geometric series. The person speaking is confused about the introduction of z0 and is asking for clarification on the process. They also mention the Taylor expansion of 1/(1-x) as a related concept.
  • #1
ThereIam
65
0
Hi all,

I am reading through Riley, Hobson, and Bence's Mathematical Methods for Phyisics and Engineering, and on page 854 of my edition they describe (I am replacing variables for ease of typing)

"expanding 1/(a-z) in (z-z0)/(a-z0) as a geometric series 1/(a-z0)*Sum[((z-z0)/(a-z0))^n] for n = 0 to infinity."

I looked up the Taylor expansion of 1/(1-x) and see that it has a geometric form (Sum[x^n] also from 0 to infinity), but I do NOT see how or why they are introducing z0. Can someone explain explicitly what's going on here?

Much thanks!
 
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  • #2
It looks like the authors wanted the students to learn how to get Taylor series around points other than 0.
 

Related to Understanding an expansion into a geometric series

1. What is a geometric series?

A geometric series is a series of numbers in which each term is obtained by multiplying the previous term by a constant ratio. For example, the series 1, 2, 4, 8, 16, ... is a geometric series with a common ratio of 2.

2. How do you find the sum of a geometric series?

The sum of a geometric series can be found using the formula S = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. Alternatively, you can also use the formula S = a / (1 - r), where a is the first term and r is the common ratio, if the series is infinite.

3. What is the difference between a finite and an infinite geometric series?

A finite geometric series has a limited number of terms, while an infinite geometric series has an infinite number of terms. The sum of a finite geometric series can be calculated using the formula S = a * (1 - r^n) / (1 - r), while the sum of an infinite geometric series can be calculated using the formula S = a / (1 - r).

4. How can geometric series be used in real life?

Geometric series can be used to model exponential growth or decay in various real-life scenarios, such as population growth, compound interest, and radioactive decay. They can also be used in financial calculations, such as calculating the future value of an investment.

5. What is the relationship between geometric series and geometric sequences?

A geometric series is the sum of a geometric sequence, where a geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant ratio. In other words, a geometric series is the collection of all the terms in a geometric sequence added together.

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