Defining y and Exploring Properties of Power Series

In summary, the conversation discusses the possibility of defining a new variable y and using it to simplify the power series and apply known convergence properties. It is suggested to write the series as a geometric series with a common ratio of x^2/9 for easier calculations.
  • #1
estro
241
0
[tex]\mbox {Suppose I have: } \sum_{n=1}^\infty (\frac {x} {3})^{2n} [/tex]

[tex]\mbox{Can I define } y= \frac {x} {3} [/tex]

[tex]
a_k(y) = \left\{
\begin{array}{c l}
(y)^k, & \mbox{if } k= 2n\\
\\
(0)^k, & \mbox{otherwise}
\end{array}
\right.
[/tex]

[tex] \mbox {And then use all the cool properties of power series on } \sum_{k=1}^\infty a_k(y)[/tex]

[I edited my question]
 
Last edited:
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  • #2
Of course, but the convergence properties you know are given in terms of [itex]y[/itex] then. You need to translate them back in terms of [itex]x[/itex]. Essentially, substitute [itex]y = (x/3)^{2}[/itex] everywhere.
 
Last edited:
  • #3
estro said:
[tex]\mbox {Suppose I have: } \sum_{n=1}^\infty (\frac {x} {3})^{2n} [/tex]

[tex]\mbox{Can I define } y= \frac {x} {3} [/tex]

[tex]
a_k(y) = \left\{
\begin{array}{c l}
(y)^k, & \mbox{if } k= 2n\\
\\
(0)^k, & \mbox{otherwise}
\end{array}
\right.
[/tex]

[tex] \mbox {And then use all the cool properties of power series on } \sum_{k=1}^\infty a_k(y)[/tex]

[I edited my question]
The simpler thing to do is write this as
[tex]\sum_{n=1}^\infty \left(\frac{x^2}{9}\right)^n[/tex]
so it is a geometric series with "common ratio" of [itex]x^2/9[/itex].
 
  • #4
estro said:
[tex]\mbox {Suppose I have: } \sum_{n=1}^\infty (\frac {x} {3})^{2n} [/tex]

[tex]\mbox{Can I define } y= \frac {x} {3} [/tex]

[tex]
a_k(y) = \left\{
\begin{array}{c l}
(y)^k, & \mbox{if } k= 2n\\
\\
(0)^k, & \mbox{otherwise}
\end{array}
\right.
[/tex]

[tex] \mbox {And then use all the cool properties of power series on } \sum_{k=1}^\infty a_k(y)[/tex]

[I edited my question]

Indeed you can, and that's how you can easily do problems without getting fooled like

[tex]\sum_{n=1}^{\infty} n^n z^{n^n}[/tex]
 
  • #5
Thank you guys!
 

1. What is the definition of "y" in a power series?

The variable "y" in a power series is typically used to represent the output or dependent variable in a mathematical function. It is the value that is being calculated or solved for in the power series.

2. How do you determine the properties of a power series?

The properties of a power series can be determined by examining its coefficients and exponents. These can provide information about the convergence, differentiability, and continuity of the series.

3. What are the common types of power series?

The most common types of power series are geometric series, binomial series, and Taylor series. Each of these has its own specific form and properties.

4. How do you find the radius of convergence for a power series?

The radius of convergence for a power series can be found by using the ratio test. This involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, the series will converge, and the radius of convergence can be determined from this value.

5. What are some real-world applications of power series?

Power series have many applications in mathematics, physics, and engineering. They can be used to approximate functions, solve differential equations, and model physical phenomena such as heat flow and electrical circuits. They are also used in computer graphics and signal processing.

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