Question

find the convergence on infinite sum from 0 to infinite of 2^(n+2)/3^n i know the answer is 12 but how do i get there.

Answer 1

Do you know how to find the convergence is for the sum\[\sum_{n=0}^\infty \left({2 \over 3}\right)^n\]

Answer 2

If you know how to find the convergence for that, you can easily find the convergence for your original sum. This is because\[\sum_{n=0}^\infty \left({{2^{n+2}} \over 3^n}\right) = \sum_{n=0}^\infty \left({{2^n \cdot 2^2} \over 3^n}\right) = \sum_{n=0}^\infty 4\cdot \left({2 \over 3}\right)^n = 4\cdot \sum_{n=0}^\infty \left({2 \over 3}\right)^n\]So if you find the convergence for what I first posted, multiply by 4 and you'll have the convergence you want.

Answer 3

*Hint* To find the convergence for the first one, use the fact that it's an infinite geometric sequence.

Answer 4

i can see it now ,thanks great help.