Answer Limit of a Series: Sum of 1+(2)^n/(3^n) from 1 to ∞

[fk378]

Homework Statement

Determine whether the series is convergent or divergent. If convergent, find its sum.

sum of [1+(2)^n] / (3^n)
from 1 to inf

Homework Equations

I know that the sum of a geometric series is 1/(1-r)

The Attempt at a Solution

The sum of a series is the limit of its partial sums.
I separate the summation into 2 parts: 1/(3^n) + (2^n)/(3^n)

I can see from this that the limits of both of these approach 0, so I conclude that the sum the series is 0.

However, my book says the answer is 5/2 and I tried to solve this a different way and got 5/2 as well. I re-wrote the separate summations as (1/3)^n + (2/3)^n and notice the ratio, r, is 1/3 and 2/3, respectively. Applying the "relevant equation" of 1/(1-r) I solve the summations and get 5/2.

However, if the sum of a series is the limit of its partial sums, why am I getting a different value for my first attempt?

 

[Dick]

You are confusing the limit of a sequence with the sum of a series. That's all.

 

[fk378]

My book says that "the sum of a series is the limit of the sequence of partial sums."

 

[Dick]

You didn't do any partial sums. You just looked at the limits of the individual terms.