What is the general formula for the sum of a telescoping series?

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In summary, the series \sum\frac{3}{n(n+3)} can be expressed as a telescoping sum using partial fraction decomposition. However, after simplifying and canceling out terms, a general formula for the sum of the series cannot be determined. Further assistance is needed to determine the convergence or divergence of the series.
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Homework Statement



Determine whether the series is convergent or divergent by expressing [tex]s_{n}[/tex] as a telescoping sum. If it is convergent, find its sum.
[tex]\sum\frac{3}{n(n+3)}[/tex]

Homework Equations


The Attempt at a Solution



Partial Fraction Decomposition: [tex]\frac{1}{n} - \frac{1}{n+3}[/tex]

Partial Sum: [tex]s_{n}= (\frac{1}{1} - \frac{1}{4}) + (\frac{1}{2} - \frac{1}{5}) + (\frac{1}{3} - \frac{1}{6}) + (\frac{1}{4} - \frac{1}{7}) + ... + (\frac{1}{n} - \frac{1}{n+3})[/tex]

From the above partial sum, I deduced that the negative term of the nth term is canceled out by the positive term in the n+4th term. However, from there, I am not able to come up with a general formula for the sum of the series. Any help would be appreciated; thanks!
 
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So which terms DON'T cancel? I count six of them for large n.
 

What is a telescoping sum?

A telescoping sum is a type of infinite series where most of the terms cancel each other out, leaving only a few terms to be added together. This allows for a simpler and more efficient way of calculating the sum.

How do you find the sum of 3/(n(n+3))?

To find the sum of this telescoping series, we must first rewrite it in a form where most of the terms will cancel out. In this case, we can use partial fraction decomposition to rewrite it as 1/n - 1/(n+3). Then, we can use the telescoping property to simplify the series to just 1/1 - 1/4, which gives us a final sum of 3/4.

What is the formula for a telescoping sum?

The formula for a telescoping sum is ∑(an - an+1), where an represents the nth term of the series. This formula is derived from the telescoping property, which states that most of the terms in a telescoping series cancel each other out, leaving only a few terms to be added together.

What is the telescoping property?

The telescoping property is a mathematical concept that states that most of the terms in a telescoping series will cancel each other out, leaving only a few terms to be added together. This allows for a simpler and more efficient way of calculating the sum of an infinite series.

How is a telescoping sum useful in mathematics?

Telescoping sums are useful in mathematics because they provide a more efficient method for calculating the sum of an infinite series. They also allow for the evaluation of certain types of series that may not be easily solved using other methods, such as geometric series or harmonic series. Additionally, they can be used in various real-life applications, such as in physics and engineering, to solve problems involving infinite series.

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