Is the Series Ʃ (1 + 2^n) / 3^n Convergent and What is Its Sum?

[Saterial]

Homework Statement

Determine whether the series is convergent or divergent. Find the sum if possible

Ʃ 1+2^n / 3^n n=1 -> infinity

Homework Equations

a/1-r

The Attempt at a Solution

I split it up so that the equation is now:

Ʃ (1/3^n) + (2/3)^n n=1 -> infinity
Ʃ (1/3^n) + Ʃ (2/3)^n n=1 -> infinity

I know that it is convergent in the second term because 2/3 < 1, how do I setup a/1-r for term 1? :S

a1=1/3 a1 = 2/3
(1/3)/(1-(1/3)) + (2/3)/(1-(2/3))
=5/2

The answer should be 3/2 on the answer sheet it says ?

 

[Mark44]

Homework Statement

Determine whether the series is convergent or divergent. Find the sum if possible

Ʃ 1+2^n / 3^n n=1 -> infinity

Homework Equations

a/1-r

The Attempt at a Solution

I split it up so that the equation is now:

Ʃ (1/3^n) + (2/3)^n n=1 -> infinity
Ʃ (1/3^n) + Ʃ (2/3)^n n=1 -> infinity

I know that it is convergent in the second term because 2/3 < 1, how do I setup a/1-r for term 1? :S

a1=1/3 a1 = 2/3
(1/3)/(1-(1/3)) + (2/3)/(1-(2/3))
=5/2

The answer should be 3/2 on the answer sheet it says ?

USE PARENTHESES!

I don't see anything wrong with your answer, but I had a hard time trying to figure out what your problem was.

This is what you wrote (fixed up in LaTeX):
$$ \sum_{n = 1}^{\infty} \left( 1 + \frac{2^n}{3^n}\right)$$

This is what I'm pretty sure you meant:
$$ \sum_{n = 1}^{\infty} \left(\frac{1 + 2^n}{3^n}\right)$$

Don't write 1+2^n / 3^n if you mean (1+2^n )/ 3^n.

 

[gustav1139]

The first sum, once you've broken them up, is not a geometric sum. Think about what that guy's doing for a little bit.

 

[Mark44]

The first sum, once you've broken them up, is not a geometric sum. Think about what that guy's doing for a little bit.

Sure it is.
The first series can be written as
$$\sum_{n = 1}^{\infty}\left(\frac{1}{3}\right)^n $$

or 1/3 + (1/3)² + (1/3)³ + ... +

 

[gustav1139]

derp. right.

 

[LCKurtz]

The first sum, once you've broken them up, is not a geometric sum. Think about what that guy's doing for a little bit.

Had I been Mark44, I would have issued you a warning or infraction for that last sentence.

 

[gustav1139]

Had I been Mark44, I would have issued you a warning or infraction for that last sentence.

I apologize. Personification is against the rules?

 

[LCKurtz]

The first sum, once you've broken them up, is not a geometric sum. Think about what that guy's doing for a little bit.

Had I been Mark44, I would have issued you a warning or infraction for that last sentence.

I apologize. Personification is against the rules?

I'm not the boss around here but I just wanted to alert you that your comment might easily be construed as violating the section below. It struck me that way anyway.

From the Rules menu at the top of the page:

Langauge and Attitude: Foul or hostile language will not be tolerated on Physics Forums. This includes profanity, obscenity, or obvious indecent language; direct personal attacks or insults; snide remarks or phrases that appear to be an attempt to "put down" another member; and other indirect attacks on a member's character or motives.

 

[gustav1139]

Ah. When I said "that guy," I was referring to the first sum in the OP's question. I guess I see how that could be misconstrued.