Determining if an infinite series converges or diverges, and finding it's sum.

[marek]

"Determine whether the given series converges or diverges. If it converges, find the sum."

The summation from n=1 to infinity of 3^(2-n)/2^n

I was able to conclude that the series converges because by comparison test, it is greater than 1/2^n. I know that the infinite series of 1/2^n converges because it is a geometric series with |r|=1/2<1.

Since this converges, I need to find the sum.

I was able to rewrite the equation as 9/[(2^n)(3^n)], but I cannot continue from here. I appreciate any help, thanks.

 

[Deleted member 4993]

"Determine whether the given series converges or diverges. If it converges, find the sum."

The summation from n=1 to infinity of 3^(2-n)/2^n

I was able to conclude that the series converges because by comparison test, it is greater than 1/2^n. I know that the infinite series of 1/2^n converges because it is a geometric series with |r|=1/2<1.

Since this converges, I need to find the sum.

I was able to rewrite the equation as 9/[(2^n)(3^n)], but I cannot continue from here. I appreciate any help, thanks.

= \(\displaystyle \dfrac{9}{6^n}\)

Do you know how to calculate the sum of the geometric series \(\displaystyle \dfrac{1}{a^n}\)?

 

[marek]

Yes, thank you for your help, I was over-thinking the problem.