MHB Determining if the sequence convergers or diverges(II)

[shamieh]

I'm confused on how they are getting their result...

Determine if the sequence converges or diverges, if it converges, find the limit...

$$\frac{n^2}{2n - 1} - \frac{n^2}{2n + 1}$$

So I started plugging in from 1 because it looks like they want me to do something with a telescoping series and I got:

(1 - 1/3) + (4/3 - 4/5) + (9/5 - 9/7) + ... (Which really got me no where?)

Then I tried to just take the limit as n -> infinity using l'opitals and ended up with $$n - n$$ which got me no where as well...So I'm really confused on how they know that it converges to $$\frac{1}{2}$$

 

[Prove It]

I'm confused on how they are getting their result...

Determine if the sequence converges or diverges, if it converges, find the limit...

$$\frac{n^2}{2n - 1} - \frac{n^2}{2n + 1}$$

So I started plugging in from 1 because it looks like they want me to do something with a telescoping series and I got:

(1 - 1/3) + (4/3 - 4/5) + (9/5 - 9/7) + ... (Which really got me no where?)

Then I tried to just take the limit as n -> infinity using l'opitals and ended up with $$n - n$$ which got me no where as well...So I'm really confused on how they know that it converges to $$\frac{1}{2}$$

What you've been given is NOT a series, so why would you try to see if this non-existent series is telescopic?

The right thing to do with this sequence is to take the limit as n approaches infinity.

$\displaystyle \begin{align*} \frac{n^2}{2n -1} - \frac{n^2}{2n + 1} &= \frac{n^2 \left( 2n + 1 \right) - n^2 \left( 2n - 1 \right) }{ \left( 2n -1 \right) \left( 2n + 1 \right) } \\ &= \frac{2n^3 + n^2 - 2n^3 + n^2}{4n^2 - 1} \\ &= \frac{2n^2}{4n^2 - 1} \\ &= \frac{2}{4 - \frac{1}{n^2} } \\ &\to \frac{2}{4 - 0} \textrm{ as } n \to \infty \\ &= \frac{1}{2} \end{align*}$

So the sequence converges to 1/2.

 

[shamieh]

What you've been given is NOT a series, so why would you try to see if this non-existent series is telescopic?So the series converges to 1/2.

Yes you are correct I am an idiot. Idk why I tried that. I'm assuming you mean the sequence converges though right? Just a typo? Again, thank you for the explanation. Helped me so much!

 

[Prove It]

Yes that was a typo, I fixed it now :)