Sum of a series question

In summary, to find the sum of the series \sum_{k=0}^\infty \frac{1}{(k+1)(k+3)}, one can rewrite the terms as fractions and then group them to get a simplified expression. However, this approach leads to incorrect cancellations and a wrong conclusion. Instead, it is necessary to carefully cancel out terms with opposite signs in order to correctly determine that the series converges to 1.
  • #1
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Homework Statement



Find the sum of the series.

[tex] \sum_{k=0}^\infty \frac{1}{(k+1)(k+3)} [/tex]


Homework Equations





The Attempt at a Solution


[tex]
= \frac{1}{1\cdot3} + \frac{1}{2\cdot4} + \frac{1}{3\cdot5} + ... + \frac{1}{(n+1)\cdot(n+3)}
[/tex]

[tex]
= \frac{1}{2} [(1-\frac{1}{3}) + (\frac{1}{2} - \frac{1}{4}) + (\frac{1}{3} - \frac{1}{5}) + ... + (\frac{1}{(n+1)} - \frac{1}{(n+3)})
[/tex]

[tex]
= \frac{1}{2}[ 1 + (\frac{1}{2} + \frac{1}{3} + \frac{1}{n+1}) - (\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... + \frac{1}{n+3}) ]
[/tex]

So here
[tex]
(\frac{1}{2} + \frac{1}{3} + \frac{1}{n+1}) \to 1
[/tex]

[tex]
(\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... + \frac{1}{n+3}) \to 0
[/tex]

Then the whole thing sums to 1?
 
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  • #2
Ok, things went very bad from the second to the third line in your argument; why don't you try to cancel the terms with opposite signs, instead of grouping them?
 

1. What is the sum of a series?

The sum of a series is the total value obtained by adding all the terms in a sequence together.

2. How do you find the sum of a series?

To find the sum of a series, you can use various mathematical methods such as the arithmetic or geometric series formula, or the sum of an infinite geometric series formula.

3. What is the difference between a finite and an infinite series?

A finite series has a limited number of terms, while an infinite series has an unlimited number of terms. The sum of a finite series can be calculated, but the sum of an infinite series may not have a finite value.

4. Can the sum of a divergent series be defined?

No, the sum of a divergent series cannot be defined as it does not converge to a finite value. This means that the terms in the series increase without bounds, making it impossible to calculate the sum.

5. How is the sum of a series used in real life?

The concept of sum of a series is used in various fields such as finance, engineering, and physics. It is used to calculate the total value of investments, determine the total force on an object, and find total distance traveled, among other applications.

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