# I know it is convergent by I cannot determine why

• golb0016
In summary, the series 0 to infinity sum of 6/(4n-1)-6/(4n+3) is convergent, as shown by using algebra to demonstrate that it is a telescoping series and by comparing it to a similar series using the comparison test.

## Homework Statement

0 to infinity sum of 6/(4n-1)-6/(4n+3)
Determine if the series is convergent or divergent.

## The Attempt at a Solution

I know it is convergent by I cannot determine why.

Did you try to see if it might be a telescoping series?

With some algebra, you should be able to show that the series converges. For example, with the particular series in question we can show that:

$$\frac{6}{4n -1} - \frac{6}{4n + 3} = 6 \left(\frac{(4n + 3) - (4n - 1)}{(4n + 3)(4n - 1)}\right) = 6\left(\frac{4}{(4n + 3)(4n - 1)}\right)$$

Using a little bit more algebra, you should easily be able to determine that the series is convergent.

Edit: Oops, somebody got here first. Sorry Dick!

jgens said:
With some algebra, you should be able to show that the series converges. For example, with the particular series in question we can show that:

$$\frac{6}{4n -1} - \frac{6}{4n + 3} = 6 \left(\frac{(4n + 3) - (4n - 1)}{(4n + 3)(4n - 1)}\right) = 6\left(\frac{4}{(4n + 3)(4n - 1)}\right)$$

Using a little bit more algebra, you should easily be able to determine that the series is convergent.

Edit: Oops, somebody got here first. Sorry Dick!

Apologies never necessarily. Besides, you showed how a similar series would converge even if it doesn't telescope using a comparison test. That's a different answer.

## 1. How can I tell if a series is convergent?

If a series has a limit as n approaches infinity, it is convergent. This means that as the terms of the series continue to be added, their sum will approach a finite number.

## 2. What is the difference between a convergent and a divergent series?

A convergent series has a finite sum as n approaches infinity, whereas a divergent series does not have a finite sum and either approaches infinity or oscillates between values.

## 3. Can you provide an example of a convergent series?

One example of a convergent series is the geometric series 1 + 1/2 + 1/4 + 1/8 + ... = 2, as the terms get smaller and approach 0, the sum of the series approaches 2.

## 4. How do you determine if a series is convergent by the comparison test?

The comparison test states that if a series is less than or equal to a convergent series, then the original series is also convergent. To use this test, you must find a convergent series that is larger than the original series.

## 5. What other methods can be used to determine if a series is convergent?

Other methods include the ratio test, the root test, and the integral test. These tests involve comparing the given series to a known series or using calculus to determine the convergence of the series.