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# Series and convergence Watch

1. Hi

I'm utterly stumped by this:

Find the exact range of x for which the sum to infinity (from n = 1) of (x^n)/( (4^n)n) converges.

Any pointers would be very much appreciated.

2. (Original post by *Georgine*)
Hi

I'm utterly stumped by this:

Find the exact range of x for which the sum to infinity (from n = 1) of (x^n)/( (4^n)n) converges.

Any pointers would be very much appreciated.

If you have s serioes of

then the radius of convergence (R) around 'A' is

In your example A=0, At the limit you can use any criteria of convergence,
for example the ratio of
Note that
3. (Original post by ztibor)
If you have s serioes of

then the radius of convergence (R) around 'A' is

In your example A=0, At the limit you can use any criteria of convergence,
for example the ratio of
Note that
I really do not understand this. How does the ratio give me the exact values of x? And what would be the a (little n) in my case?
4. You are considering , with .

By the ratio test:

- if then the series converges.

- if then the series diverges.

- if , then you can't tell anything and need to examine this case separately.

Use the first bullet point to find what range x has to be in to make the ratio less than 1 - you know that the series converges for all x in this range.

Then examine the case where the ratio = 1 directly i.e. find the value of x that makes the ratio 1, and see if you think the series converges/diverges for this particular value of x.
5. (Original post by Daniel Freedman)
Then examine the case where the ratio = 1 directly i.e. find the value of x that makes the ratio 1, and see if you think the series converges/diverges for this particular value of x.
Given that there were no restrictions on where x lies, I would say 'values of x that make the ratio 1'.
6. (Original post by ljfrugn)
Given that there were no restrictions on where x lies, I would say 'values of x that make the ratio 1'.
Indeed. Thanks
7. (Original post by *Georgine*)
I really do not understand this. How does the ratio give me the exact values of x? And what would be the a (little n) in my case?
Not the ratio gives but the limit of ratio does a value which maybe finite or infinite or even 0. This value is the reciprotial of R radius,
For example if the limit is L then

and the series will convergent if

so

With the nth root test

with the ratio

Calculating the limit with R you will get the range of convergence and consider A=0 (convergence around zero)

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