Find Power Series Representation for $g$: Interval of Convergence

In summary, to find the power series representation for $g$ centered at 0, we can either differentiate or integrate the power series for $f$, possibly more than once. The interval of convergence for the new series is unknown without more information about the relationship between $f$ and $g$.
  • #1
karush
Gold Member
MHB
3,269
5
$\textrm{a. find the power series representation for $g$ centered at 0 by differentiation}\\$
$\textrm{ or Integrating the power series for $f$ perhaps more than once}$
\begin{align*}\displaystyle
f(x)&=\frac{1}{1-3x} \\
&=\sum_{k=1}^{\infty}
\end{align*}
$\textsf{b. Give interval of convergence of the new series } $

just reviewing but ? on this one
 
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  • #2
How are the the functions $f$ and $g$ related?
 
  • #3
skeeter said:
How are the the functions $f$ and $g$ related?
do you suggest this:
$$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$
 
  • #4
karush said:
do you suggest this:
$$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$

The information regarding how $f$ and $g$ are related is missing...without that, we cannot help. :D
 
  • #5
karush said:
do you suggest this:
$$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$

I didn't suggest anything ... I don't know the relationship between $f$ and $g$ because you have not provided that essential piece of information.
 
  • #6
it was from math lab which I don't have acess to anymore. so g probably was noted there..

sorry I just drop the problem
 

FAQ: Find Power Series Representation for $g$: Interval of Convergence

What is a power series representation?

A power series representation is an infinite series that can be used to approximate a function, by expressing it as a sum of powers of a variable. It is written in the form of ∑ an (x-c)n, where an represents the coefficients and c is the center of the series.

How do you find the power series representation of a function?

To find the power series representation of a function, we need to first identify the center of the series, c. Then, we can use the formula ∑ an (x-c)n to find the coefficients, an. These coefficients can be found by differentiating the function and evaluating it at the center, c.

What is the interval of convergence for a power series representation?

The interval of convergence is the range of values for the variable, x, for which the power series converges and accurately approximates the function. It is typically expressed in terms of the distance from the center, c, to the boundary of the interval of convergence.

How do you determine the interval of convergence for a power series representation?

To determine the interval of convergence, we can use the ratio test or the root test. These tests involve taking the limit of the absolute value of the ratio or the root of the coefficients, an, and comparing it to a specific value. If the limit is less than the value, the power series converges, and the interval of convergence can be determined from the distance between the center, c, and the boundary.

Why is it important to find the interval of convergence for a power series representation?

The interval of convergence is important because it tells us the range of values for which the power series will accurately approximate the function. It also helps us determine if the series can be used to evaluate the function at specific values, or if it is only an approximation. Additionally, it helps us identify any potential issues or limitations with the power series representation.

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