Finding the sum of a Series that is converge or Diverge

[Physicsnoob90]

Homework Statement

Determine whether the series is either Converge or Diverge, if it's convergent, find its sum

∑ from n=1 to ∞ of (1+2^n)/(3^n)

Homework Equations

The Attempt at a Solution

Steps:
1) i replaced the (1+2^n) to just (2^n) and my equation behaves like ∑ (2/3)^n which the Radius (2/3) < 1. So it's Convergent.
2) I found that my leading term was 2/3 by n=1
3) i plug everything into my Geometric series formula (2/3)/(1-(2/3), thus giving me an answer of 2.

Problem: the Books answer was 5/2. Can anyone help me out? Thanks!

 

[Dick]

Homework Statement

Determine whether the series is either Converge or Diverge, if it's convergent, find its sum

∑ from n=1 to ∞ of (1+2^n)/(3^n)

Homework Equations

The Attempt at a Solution

Steps:
1) i replaced the (1+2^n) to just (2^n) and my equation behaves like ∑ (2/3)^n which the Radius (2/3) < 1. So it's Convergent.
2) I found that my leading term was 2/3 by n=1
3) i plug everything into my Geometric series formula (2/3)/(1-(2/3), thus giving me an answer of 2.

Problem: the Books answer was 5/2. Can anyone help me out? Thanks!

You can't throw away the 1 if you expect to get an exact sum. Put it back in.

 

[Mugged]

Yes, keep the 1 inside. I understand your idea for breaking up the sum, but you forgot about the other term.

Your split idea is good: (1+2^n)/(3^n) = 1/3^n + 2^n/3^n.

Use your formula for both these terms. Hint: 1 = 1^n.