Converges/Diverges Problem

[thatguy47]

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



How would you go about solving this? It's different than the rest of the problems I've been doing and there's no examples like this in my notes.

 

[galactus]

Rewrite it as:

\(\displaystyle \frac{1}{3}\cdot a_{n}\)

\(\displaystyle \frac{1}{3}\cdot \frac{2^{n}}{3^{n}}\)

\(\displaystyle \frac{1}{3}\cdot (\frac{2}{3})^{n}\)

Now, can you see what it does?. See the limit?.

List out some terms if it helps. If the even numbered terms and the odd numbered terms both converge to a limit than so does the sequence.

 

[thatguy47]

Could you just explain where the 1/3 comes from?

 

[galactus]

Uh oh. You don't see that?.

Doesn't \(\displaystyle 3^{n+1}=3\cdot 3^{n}\)?.

 

[thatguy47]

Ya...I didn't see that, I completely forgot learning that. Thanks.