Evaluate an Infinite Series in Closed Form

In summary, evaluating an infinite series in closed form means expressing it as the sum of a finite number of well-understood functions. An example of this is the series \sum_{n=0}^{\infty}x^n=\frac{1}{1-x}, where the left hand side is an infinite series and the right hand side is in closed form. It is important to note that this does not involve solving the series, but rather finding a simplified, finite representation for it.
  • #1
avocadogirl
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My question is one of vocabulary. What does it mean to evaluate an infinite series in closed form?

If I have a Series: [tex]\Sigma[/tex] 1/ (N2), as N goes from 1 to infinity.

This is similar to a test question I'm working on so, I DO NOT want to know how to solve it, I just want to know exactly what is meant by evaluating an infinite series in closed form.
 
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  • #2
"Expressed in closed form" means "expressed as sum of finitely many well-understood functions".

Here's an example. Let [itex]|x|<1[/itex]. Then,

[tex]\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}[/tex]

The expression on the left hand side is an infinite series, but the expression on the right is in closed form.
 
  • #3
Thank You!
 

1. What is an infinite series?

An infinite series is a sum of an infinite number of terms. It is represented by the notation ∑ an, where an is the nth term in the series.

2. What does it mean to evaluate an infinite series in closed form?

Evaluating an infinite series in closed form means finding a formula or expression that represents the sum of all the terms in the series. This allows us to find the exact value of the series without needing to add an infinite number of terms.

3. How do you evaluate an infinite series in closed form?

To evaluate an infinite series in closed form, we can use various mathematical techniques such as geometric series, telescoping series, and power series. These techniques involve finding a pattern in the terms of the series and using algebraic manipulations to simplify the sum.

4. Why is it important to evaluate an infinite series in closed form?

Evaluating an infinite series in closed form allows us to find the exact value of the series, which can be useful in many practical applications. It also helps us understand the behavior and properties of infinite series, which are important in fields such as physics, engineering, and economics.

5. What are some common examples of infinite series that can be evaluated in closed form?

Some common examples of infinite series that can be evaluated in closed form include geometric series, harmonic series, alternating series, and power series. These types of series have well-known formulas or expressions that represent their sums.

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