Question

Find all values of x for which Σ n=1to ∞( (x-2)/3)^n is a convergent geometric series, then express the sum of the series as a function of x.

Answer 1

Looks horrible... :3

Answer 2

calc 2 baby hahah

Answer 3

Thankfully, I'm here to watch you :)

Answer 4

Here, I'll even write it for you
\[\huge \sum_{n=1}^\infty\left(\frac{x-2}3\right)^n \]

Answer 5

Now, when is a geometric series convergent ^.^

Answer 6

infininity over infinity

Answer 7

?

Answer 8

No, you misunderstand... I just asked when a geometric series is convergent...?

Answer 9

If |r| < 1

Answer 10

That's right :)
Now what is r in this specific series?

Answer 11

(x-2)/3?

Answer 12

Right again :D
So you want
\[\huge \left|\frac{x-2}{3}\right|<1\]Now solve :>

Answer 13

x<5

Answer 14

and?

Answer 15

x>-1

Answer 16

Correct :)
So your range for x is just
(-1 , 5)

Answer 17

now how can i express is at a sum of the series? do i do that a/1-r

Answer 18

Yeah, and as usual, replace r properly...
\[\huge r = \frac{x-2}{3}\]

Answer 19

so 1/1-((x-2)/3)

Answer 20

Yup, and simplify as desired ^.^

Answer 21

then what

Answer 22

simplify it, and I'll tell you what then :)

Answer 23

3/1-x

Answer 24

And that's actually it :)

Answer 25

You have just expressed the sum of the series as a function of x.

Answer 26

wow thank you so much you were a huge help :) you know your stuff

Answer 27

That I do ^_^

Answer 28

prof or past calc 2 student?

Answer 29

Uhh.. neither :)

Answer 30

10th grader, actually ^_^
(You just had to ask :> )

Answer 31

no way

Answer 32

your one smart 10th grader

Answer 33

Well, I guess 10th grader is more believable than "ageless kid from the second to the right and straight on till morning"

But I am in 10th grade, though :)

Answer 34

how d you know calc two stuff

Answer 35

Books. Too much time on my hands :)

Answer 36

dang well thanks

Answer 37

Any time (well, not really :> )

Answer 38

^^^ this guy

Answer 39

Cheerio ^.^

Answer 40

so you were not completely right but i fixed the problem