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# Interval of Convergence for acrtan(x) series watch

1. So I derived the general formula for each term in the power series of arctan(x) :

I then used the ratio test and simplified to get :

Giving just lx^2l which means the series only converges for :

-1<x^2 <1

At this point I'm stuck on what to do. How do I proceed ? I cant simply take a square root of each side....
2. (Original post by Ari Ben Canaan)
So I derived the general formula for each term in the power series of arctan(x) :

I then used the ratio test and simplified to get :

Giving just lx^2l which means the series only converges for :

-1<x^2 <1

At this point I'm stuck on what to do. How do I proceed ? I cant simply take a square root of each side....
There is not x^2 at the limit when you calculate the radius.
The series is given around 0
the range of convergence |x-0|<1
3. (Original post by Ari Ben Canaan)
Giving just lx^2l which means the series only converges for :

-1<x^2 <1

At this point I'm stuck on what to do. How do I proceed ? I cant simply take a square root of each side....
Why not?

Spoiler:
Show
You might also want to observe that it is always true that 0<= x^2.
4. (Original post by DFranklin)
Why not?

Spoiler:
Show
You might also want to observe that it is always true that 0<= x^2.
Sorry, I meant to say taking the square root would mean getting i on the LHS.

Hmmm, so essentially the interval of convergence is just 0<x<1 ?
5. (Original post by Ari Ben Canaan)
Hmmm, so essentially the interval of convergence is just 0<x<1 ?
You're forgetting the "other half"
6. (Original post by Lord of the Flies)
You're forgetting the "other half"
I don't quite follow...
7. (Original post by ztibor)
There is not x^2 at the limit when you calculate the radius.
The series is given around 0
the range of convergence |x-0|<1
What do you mean there is no x^2 ? Of course there is....
8. (Original post by Ari Ben Canaan)
I don't quite follow...
I was hinting at negative values of x (also, don't forget to check endpoints!).
9. (Original post by Lord of the Flies)
I was hinting at negative values of x (also, don't forget to check endpoints!).
-1<x<1 ?

Mu question as to how to deal with the x^2 still hasn't been answered or at least I haven't understood it yet. Could you please just spell out for me ?
10. (Original post by Ari Ben Canaan)
-1<x<1 ?

Mu question as to how to deal with the x^2 still hasn't been answered or at least I haven't understood it yet. Could you please just spell out for me ?
There's nothing to deal with really:

Since the ratio test is inconclusive when you need to check if the series converges at these values (it does).

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Updated: March 28, 2013
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