What is the correct power series for $g(t)$?

In summary, the power series of a function around a point \(t_0\) is given by \(g(t) = \sum_{n=0}^{\infty} \frac{(t-t_0)^n g^{(n)}(t_0)}{n!}\), where \(g^{(n)}(t_0)\) represents the nth derivative evaluated at \(t_0\). This is known as the Taylor power series.
  • #1
Dustinsfl
2,281
5
What would be the power series of $g(t)$?

$$
g(t) = \sum_{n=0}^{\infty}\frac{g(t)}{n!}
$$

This?
 
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  • #2
?? What EXACTLY are you trying to do? I'm guessing the recursive definition you have suggested is not a good way to go.
 
  • #3
dwsmith said:
What would be the power series of $g(t)$?

$$
g(t) = \sum_{n=0}^{\infty}\frac{g(t)}{n!}
$$

This?

There is a mistake in your post, as posted there is no such function other than the zero function.

CB
 
  • #4
The power series of a function around a point \( t_0 \) is $$ g(t) = \sum_{n=0}^{\infty} \frac{(t-t_0)^n g^{(n)}(t_0)}{n!} .$$ Note that the \( g^{(n)}(t_0) \) denotes the derivative evaluated at \( t_0 \), where \( g^{(0)}(t_0) = g(t_0) \).

Does this help you?

Edit: Sorry, to be specific this is the Taylor power series.
 
  • #5
There was a mistake in what I was reading, i.e. is read it wrong. I see what my issue was so the question I asked was wrong.
 

FAQ: What is the correct power series for $g(t)$?

What is a power series?

A power series is a mathematical series that is used to represent a function as an infinite sum of terms. It is typically written in the form of a polynomial with infinitely many terms, where each term has a variable raised to a different exponent.

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

The radius of convergence for a power series is found by applying the ratio test. This involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If this limit is less than 1, the series converges, and the radius of convergence is equal to the distance from the center of the series to the nearest point at which the series diverges.

What is the difference between a power series and a Taylor series?

A power series is a specific type of series that represents a function, while a Taylor series is a type of power series that is centered at a specific point. A Taylor series includes terms that represent the derivatives of the function at that point, while a general power series does not have this restriction.

What is the purpose of using a power series?

Power series are useful in many mathematical applications, such as approximating functions, solving differential equations, and evaluating integrals. They allow for complex functions to be represented in a more manageable and understandable form.

Can any function be represented by a power series?

No, not all functions can be represented by a power series. The function must be analytic, meaning that it can be represented by a convergent power series in some interval. Functions that are not analytic, such as step functions or functions with discontinuities, cannot be represented by a power series.

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