Question

Power Series question?

Answer 1

\[\sum_{}^{}\frac{ (-1)^k(x)^k }{ 3^k k}\]

Answer 2

We took the limit after doing the ratio test and found it to be (1/3)

Answer 3

But now we are stuck

Answer 4

@MARC_D @mww

Answer 5

@UnkleRhaukus

Answer 6

@Callisto

Answer 7

@pooja195

Answer 8

Should be -1/3

Answer 9

Are you asked to evaluate:
\[\sum_{k=1}^{\infty} \frac{(-1)^k x^k}{3^k \cdot k}\]
is that the question?
Or is the question something else?

Answer 10

Using the power series, yes

Answer 11

kinda confused...
are you asked to find radius of convergence ?

Answer 12

assuming the actual question
was to find the radius of convergence

it should look like this:
\[A_k=(\frac{-1}{3})^k x^k \cdot \frac{1}{k} \\ \text{ We want to solve: } \lim_{k \rightarrow \infty} | \frac{A_{k+1}}{A_k}|<1 \text{ for } x \]

Answer 13

well that would be the interval
but we can still determine the radius of convergence from the interval

Answer 14

Yes, radius of convergence. Sorry about that.

Answer 15

What did you get when you tried to solve the above inequality ?

Answer 16

I got (-1/3) but the inequality confused me. I also didnt seperate it out like that, just did the ratio test then simplified, pulled the x outside of the limit and took the limit...(-1/3)

Answer 17

okay so you have gotten here so far:
\[\lim_{k \rightarrow \infty} |\frac{A_{k+1}}{A_k}|=|\lim_{k \rightarrow \infty} \frac{A_{k+1}}{A_k}|=|\frac{-1}{3}x|\]
So you still need that to be less than 1 in order to have convergence

Answer 18

\[|\frac{-1}{3} x|<1 \\ |\frac{-1}{3} \cdot x|<1 \\ |\frac{-1}{3} | \cdot | x| <1 \\ \frac{1}{3} |x|<1\]

Answer 19

now this is the same as |x|<3

do you know how to tell the radius from this ?

Answer 20

I think it would be from 0 to 3

Answer 21

well a radius is a length
and the length between 0 to 3 is ?

Answer 22

Ah, 3

Answer 23

yep

Answer 24

3 is the radius of convergence
good job

Answer 25

What if I needed to find the interval of convergence, would it just be (0,3)

Answer 26

And then plug my endpoints back into the series for x and evaluate each of them respectively to determine if the series converges or diverges at the endpoints?

Answer 27

|x|<3 means all the values that satisfy |x|<3
there are more values than just (0,3) that satisfy this inequality

Answer 28

|x|<3 means all the values that gives a distance less than 3 from 0

Answer 29

and numbers in (-3,0] also do this

Answer 30

|x|<3 means (-3,3)

Answer 31

Oh I see what you mean

Answer 32

you can imagine the diameter to be from x=-3 to x=3 if you want
half that will be the radius

Answer 33

So (-3, 3) would be my interval of convergence

Answer 34

yep because all the values in (-3,3) satisfy |x|<3

Answer 35

I like that way of thinking about it. The diameter

Answer 36

You guys are a great help, my test is in an hour

Answer 37

the diameter would have length 6
and the radius would be 6/2 or 3

just like circles
although they never mention diameter of convergence
that's my own term

Answer 38

Yea, I get what you mean though

Answer 39

Thanks again, thats a big help. I was stuck on that question all morning

Answer 40

it will get easier :)

Answer 41

I hope so lol