Find closed form of series SUM (nx)^(2n)

Thread starter Calculus!
Start date May 1, 2009
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Closed Form Series Sum

In summary, the closed form of the series SUM (nx)^(2n) is 1/(1-x)^3, and to find the closed form of a series, one needs to identify the pattern and use techniques such as substitution, algebraic manipulation, and geometric series formulas. The significance of the closed form includes simplifying the sum of an infinite series and aiding in understanding its behavior and properties. Common techniques used include substitution, algebraic manipulation, geometric series formulas, and calculus techniques. However, limitations may arise, such as some series not having a closed form or requiring advanced mathematical knowledge.

May 1, 2009

Calculus!

If abs x < 1 find a closed form function (i.e. f(x) = x +1) for the following series:

[tex]\sum((nx)^(2n))[/tex]

(reads: the series from n=1 to infinity of nx^(2n))

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May 1, 2009

eok20

Try to first find an equation for sum n*x^n = x*sum n*x^(n-1) using integration/differentiation.

FAQ: Find closed form of series SUM (nx)^(2n)

What is the closed form of the series SUM (nx)^(2n)?

The closed form of the series SUM (nx)^(2n) is 1/(1-x)^3, where x is the ratio of the common difference in the series.

How do you find the closed form of a series?

To find the closed form of a series, you need to identify the pattern in the terms of the series and use mathematical techniques such as substitution, algebraic manipulation, and geometric series formulas to simplify and find a general formula for the sum of the series.

What is the significance of the closed form of a series?

The closed form of a series allows us to find the sum of an infinite series without having to add each term individually. It also helps in understanding the behavior and properties of a series, and it can be used in various mathematical and scientific applications.

What are the common techniques used to find the closed form of a series?

The common techniques used to find the closed form of a series include substitution, algebraic manipulation, geometric series formulas, and techniques from calculus such as integration and differentiation.

Are there any limitations to finding the closed form of a series?

Yes, there are limitations to finding the closed form of a series. Some series may not have a closed form, and in those cases, other mathematical techniques such as numerical methods may be used to approximate the sum. Additionally, finding the closed form of a series may require advanced mathematical knowledge and may not be possible for all individuals.