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# Radius of convergence of the power series watch

1. I have got to the point where I have

| k(1-x) |
-----------
(k+1)

Where exactly do I go from here to find the radius of convergence?
2. (Original post by Bruce Harrisface)

I have got to the point where I have

| k(1-x) |
-----------
(k+1)

Where exactly do I go from here to find the radius of convergence?

I am a bit busy to do the question but assuming what you are writing is correct, then you want the limit of this as k tends to infinity to be strictly less than 1

This will give you the range of x

(PS If this is what year 9s do these days, then the Government reforms in education are clearly working)
3. (Original post by Bruce Harrisface)

I have got to the point where I have

| k(1-x) |
-----------
(k+1)

Where exactly do I go from here to find the radius of convergence?
By the ratio test, a series with terms converges (absolutely) if and diverges if the same limit is > 1.

In your case, you are considering - note that the -1 terms mod to 1, so I haven't shown them.

Bear in mind that is some arbitrary real number. So your job is to:

1. find the limit of the expression above - the term doesn't take part in the limiting process though - it's just some arbitrary number.
2. solve an inequality imposed by the ratio test to find the set of values of which satisfy it.

In addition, the ratio test says nothing if the limit *is* 1 - you will have to consider these cases separately to see if the end points 1 and -1 also give convergence.

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Updated: January 21, 2016
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