Summing the Series: $\sum (1+2^n)/3^n$

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Homework Statement

$$\sum (1+2^n)/3^n$$

Find its sum

Homework Equations

The Attempt at a Solution

I don't know how to figure this one out. I took the limit and it was zero so I know it converges. I don't know where to start on this one. The answer is 5/2.

 

[TMFKAN64]

Write this as the sum of two summations.

 

[HallsofIvy]

More precisely, the sum of two geometric series.

 

[Gib Z]

And just because the the terms approach zero, the series always convergent. Think of the harmonic series.

 

[HallsofIvy]

Typo? I'm quite sure Gib Z meant "just because the terms approach zero, the series is not always convergent". Yes, this particular series converges but your statement " I took the limit (of the sequence of terms) and it was zero so I know it converges" does not follow.
The fact that the limit of the sequence of terms goes to zero is necessary for convergence but not sufficient.

 

[Gib Z]

Typo indeed :( Was in a rush for that post lol.