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

• kahwawashay1
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.
kahwawashay1
what does it mean if y(x)=$\sum$$^{∞}_{n=0}$ an(x-x0)n
and y(0)=1 ?
does this mean that x0=0 and a0=1 ?

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)=$\sum$$^{∞}_{n=0}$ 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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