Calculate the sum of the series

[shemer77]

so i am having trouble calculating the sum of the series of
(1+2^x)/(3^x) from 1 to infinity
i rearranged it and made a geometric series where r =2/3 and got that it converges toward 2. however the answer is 2.5? This problem shouldn't need any tests as its it a section of the book that hasant taught any of the tests, except for geometric so what am I doing wrong. I have been stuck on this for hours now? :(

 

[SammyS]

Let's see details for how you went about getting your result.

 

[Bohrok]

so i am having trouble calculating the sum of the series of
(1+2^x)/(3^x) from 1 to infinity
i rearranged it and made a geometric series where r =2/3 and got that it converges toward 2. however the answer is 2.5? This problem shouldn't need any tests as its it a section of the book that hasant taught any of the tests, except for geometric so what am I doing wrong. I have been stuck on this for hours now? :(

You forgot about the 1 in the numerator:

\sum \frac{1 + 2^x}{3^x} = \sum \left(\frac{1}{3^x} + \frac{2^x}{3^x} \right) = \sum \frac{1}{3^x} + \sum \left(\frac{2}{3}\right)^x = ?

 

[nickalh]

Borhok has the right method again.
Shemer77, would you let us know if you've gotten the right answer from Bohrok's information or still have further questions? Thanks.