Question

Anybody good with sequences and series?

Answer 1

What is the sum?
\[\sum_{n=1}^{\infty}(-1)^n (\frac{2}{3})^{n-2}\]

Answer 2

@lgbasallote help him!

Answer 3

Well, you know... please. :)

Answer 4

Really, I'm just wondering if there is a formula I can use. Can I use the typical geometric a/(1-r) for an alternating infinite series like this? Or is the formula a little different because it is alternating?

Answer 5

Can you break it up into negative and positive terms, use the geometric model for each one, and then add the results?

Answer 6

I bet that would work. I'll give it a shot.

Answer 7

Perhaps I could also multiply the (-1) with the (2/3) because the way the exponents effect the numbers, the series would be the same.

Answer 8

Yikes! Maybe not... :)

Answer 9

I'll try the positive sum + negative sum idea. Thanks!!

Answer 10

The negative terms would be:
Starting term (-2/3) common ratio 4/9
Except all multiplied by (-3/2)

The positive terms would be
Starting term 1 common ration 4/9

Right?

Answer 11

The negative terms gave me -1.8
and the positive terms gave me 1.8
So I'm pretty sure this series telescopes to 0.

The other thing is that there's probably some fancy way to rearrange the terms so that they cancel out.

Answer 12

I see the negative terms starting at -3/2 and the positive terms starting at 1 with a common ration of 2/3

Answer 13

See well... I don't think it makes sense to consider -3/2 to be the starting value. What is the second negative term of the sequence?

Answer 14

-2/3

Answer 15

The negative terms:
(-3/2) (-2/3) (-8/27) (-32/243)
Don't you agree?

Answer 16

I do.

Answer 17

Okay, actually it should work to consider (-3/2) the first term. What is the common ratio?

Answer 18

The common ratio is 2/3

Answer 19

(-2/3)*(2/3) = (-8/27)?

Answer 20

(-3/2)*(2/3) = (-2/3)? The math there isn't working out...

Answer 21

Oh - would the common ration need to change because we are removing the negative terms when we sum the positive values only?

Answer 22

ration = ratio

Answer 23

(2/3) was the common ratio for the original sequence, but for the negatives only, (2/3) is getting multiplies twice to get to the next negative, so the new common ratio is
*(2/3)*(2/3) = *(4/9)

Answer 24

You're right! And I can see how that would work with the powers too... ok, I'm on board. :)

Answer 25

Cool. But I agree with you that (-3/2) is the starting value for the negative sequence. And that changes the answer that I get for the sum of that sequence.

Answer 26

http://www.wolframalpha.com/input/?i=%28-3%2F2%29%2F%281-%284%2F9%29%29+%2B+1%2F%281-%284%2F9%29%29

Answer 27

Yep! That's it!!

Answer 28

Thank you so much!!

Answer 29

My pleasure =D