Finding a Closed Form from a Power Series

In summary, a power series is an infinite series with a constant and a sequence of coefficients, while a closed form is an expression that can be written in a finite number of terms. To find a closed form from a power series, various methods can be used, such as substitution, differentiation, and integration. However, not all power series can be expressed in closed form. Applications of finding a closed form from a power series can be seen in various fields such as physics, engineering, and finance.
  • #1
Frillth
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Homework Statement



I have f(x) = the series x^k/[(k-1)k] summed from x=2 to infinity, and I need to find its closed form. Hint: What is the derivative of f(x)

Homework Equations



None.

The Attempt at a Solution



To start this problem, I took the derivative of the series, which makes the general term x^(k-1)/(k-1), which is x + x^2/2 + x^3/3... How do I get a function from this? I know this series looks really familiar, but I can't seem to remember where I've seen it before.
 
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  • #2
Take the derivative one more time.
 

Related to Finding a Closed Form from a Power Series

1. What is a power series?

A power series is an infinite series of the form ∑n=0∞ an(x-a)n, where a is a constant and an is a sequence of coefficients.

2. How do you find a closed form from a power series?

To find a closed form from a power series, you can use various methods such as substitution, differentiation, and integration. The goal is to manipulate the series in a way that it can be expressed as a familiar function, such as a polynomial or a trigonometric function.

3. What is the difference between a closed form and an open form?

A closed form is an expression that can be written in a finite number of terms, while an open form is an expression that requires an infinite number of terms to be written. In other words, a closed form has a finite number of operations and can be evaluated to a specific value, whereas an open form is an infinite process and cannot be evaluated to a specific value.

4. Can all power series be expressed in closed form?

No, not all power series can be expressed in closed form. Some power series do not have a closed form, meaning they cannot be expressed as a finite number of terms. However, many commonly used power series, such as geometric series and Taylor series, do have closed forms.

5. What are some real-world applications of finding a closed form from a power series?

Finding a closed form from a power series can be used in various scientific and mathematical fields, such as physics, engineering, and finance. For example, in physics, power series are used to approximate the behavior of physical systems, and finding a closed form can provide a more precise understanding of the system. In finance, power series can be used to model the growth of investments over time, and finding a closed form can help in making financial predictions.

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