Question

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Answer 1

The usual trick is some telescoping - AFTER you are sure that it converges.

Here is the sum
1/3 + 2/9 + 3/27 + ... + n/3^n + ... = S

Pick at least part of the common factor.
1/9 + 2/27 + 3/81 + ... + n/3^(n-1) + ... = S/3

And subtract.
1/3 + 1/9 + 1/27 + ... + 1/3^n = S - S/3 = 2S/3

Are we getting anywhere?

Answer 2

I'm not really getting it

Answer 3

1/3 + 2/9 + 3/27 + ... + n/3^n + ... = S
Do you see that? I just named the total, assuming that it exists. Any trouble with that?

Answer 4

Yeah, but, how do I find an actual numerical answer?

Answer 5

No jumping ahead. You need to follow one thing at a time.

Now that we have S, we can play with it a little. With a little judgment, we can play with it in a useful way. Since the denominator increases by a factor of 3, I decided the next step would be to divide by 3.

1/3 + 2/9 + 3/27 + ... + n/3^n + ... = S
1/9 + 2/27 + 3/81 + ... + n/3^(n+1) + ... = S/3

Follow that? I inadvertently wrote n-1 above. It should have been n+1.

Answer 6

Got it, yeah

Answer 7

Or wait is the answer 2/3?

Answer 8

Jaime: Weren't there instructions with these problems?

Answer 9

There was a formula that I had to take the derivative of both sides for, so I did that, and then all it said was to use that formula to compute the series I wrote in the question.

Answer 10

Why do you keep jumping ahead. Why not understand ALL of it before you think you have the answer?

Now, we are going to subtract them.

1/3 + 2/9 + 3/27 + ... + n/3^n + ... = S
1/9 + 2/27 + 3/81 + ... + n/3^(n+1) + ... = S/3

1/3 + (2/9 - 1/9) + (3/27 - 2/27) + ... = S - S/3 = 2S/3

1/3 + 1/9 + 1/27 + 1/81 + ... = 2S/3

Are you buying this, so far? It's just arithmetic, but in a very different way from what you may have learned earlier. This time, we are performing infinitely many subtractions all at the same time.

Answer 11

Alright

Answer 12

Okay, do you recognize that thing on the left hand side?

Answer 13

Somewhat

Answer 14

What is it?

Answer 15

Never mind, I don't really

Answer 16

You SHOULD recognize a Geometric Series. First term is 1/3. Common Ratio is 1/3.

Are we ringing any bells?

Answer 17

Oh, I wasn't sure what you were asking. But yeah, I know geometric series, yes.

Answer 18

Do you know the sum of this geometric series?

Answer 19

I'm not entirely sure.

Answer 20

You're going to make me do all the work, aren't you?

Please find the sum of that geometric series.

First Term = 1/3
Common Ratio = 1/3

Go!!

Answer 21

13/27?

Answer 22

\(\dfrac{1/3}{1 - 1/3} = \dfrac{1/3}{2/3} = 1/2\)

This leaves us with \(1/2 = 2S/3\) Solve for S and you are done.