I know it is convergent by I cannot determine why

[golb0016]

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.

 

[Dick]

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

 

[jgens]

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!

 

[Dick]

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.