Power Series x=0: What Does a0 & x0 Mean?

In summary, the conversation discusses the meaning of the function y(x) which is equal to the infinite sum of an(x-x0)n, and how the initial condition y(0)=1 is related to the values of x0 and a0. It is mentioned that while x0=0 and a0=1 is a possible choice, other options may be used depending on the situation.
  • #1
kahwawashay1
96
0
what does it mean if y(x)=[itex]\sum[/itex][itex]^{∞}_{n=0}[/itex] an(x-x0)n
and y(0)=1 ?
does this mean that x0=0 and a0=1 ?
 
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  • #2
It means y(0)=1.

That x0=0 and a0=1

is one (obvious) possibility, but sometimes other choices are used.

For example we might chose a different x0 is there is a singularity near zero and we need the series to converge is a certain range. Like if there were a singularity at x=-1 and we would like a series valid on (-1,3) we might choose x0=2. Then y(0)=1 means

1=y(0)=[itex]\sum[/itex][itex]^{∞}_{n=0}[/itex] an(-2)n

A bit tricky to work with, but not too bad.
 

1. What is the purpose of the x=0 in a power series?

The x=0 in a power series indicates the point at which the series is centered around. It is often referred to as the "center" or "origin" of the series.

2. What does a0 represent in a power series?

The term a0 in a power series represents the coefficient of the constant term, or the term without any x variables. It is often called the "leading coefficient" or "initial term."

3. What is the significance of x0 in a power series?

The x0 in a power series represents the power of x that is being raised to. For example, x^0 is equivalent to 1, x^1 is equivalent to x, and so on. It helps to determine the pattern and behavior of the series.

4. How do a0 and x0 affect the overall power series?

The values of a0 and x0 can greatly impact the overall behavior and convergence of a power series. The value of a0 determines the "height" of the series, while the value of x0 determines the "width" or "radius" of the series.

5. Can a0 or x0 be negative in a power series?

Yes, both a0 and x0 can be negative in a power series. This can result in a series with a downward or outward trend, depending on the exponents of the other terms in the series.

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