Hi,
Could someone help me with these questions,I dont understand how you can find the radius of convergence:

1.f(x)=(1+4x)^3/2

State the radius of convergence of this power series

(I also had to use the General Binomial Theorem to determine the first four terms of the Taylor series at 0 for this series but I have done this)

2.g(x)=6(1+4x)^1/2

State the radius of convergence of this power series

(I also had to write down the first three terms of the Tayor series at 0 but I have done this)

2. Originally Posted by mountaindream
Hi,
Could someone help me with these questions,I dont understand how you can find the radius of convergence:

1.f(x)=(1+4x)^3/2

State the radius of convergence of this power series

(I also had to use the General Binomial Theorem to determine the first four terms of the Taylor series at 0 for this series but I have done this)

2.g(x)=6(1+4x)^1/2

State the radius of convergence of this power series

(I also had to write down the first three terms of the Tayor series at 0 but I have done this)
The first thing you need to do is get the general term of the power series for each of the given functions.

3. Originally Posted by mr fantastic
The first thing you need to do is get the general term of the power series for each of the given functions.
I dont understand what you mean-do you know how to do this?

4. Originally Posted by mountaindream
I dont understand what you mean-do you know how to do this?
Have you studied power series at all (eg. Maclaurin series, etc.)? Have you learned how to write functions in the form $\displaystyle f(x) = \sum_{n=1}^{\infty} a_n x^n$?

The functions f(x) and g(x) are not power series. So the first step is to write them in the form $\displaystyle f(x) = \sum_{n=1}^{\infty} a_n x^n$.

5. Originally Posted by mr fantastic
Have you studied power series at all (eg. Maclaurin series, etc.)? Have you learned how to write functions in the form $\displaystyle f(x) = \sum_{n=1}^{\infty} a_n x^n$?

The functions f(x) and g(x) are not power series. So the first step is to write them in the form $\displaystyle f(x) = \sum_{n=1}^{\infty} a_n x^n$.
No I have not learnt about that kind of series yet. ~It just seems it should be obvious because the question says "state" which implies it should be simple to detect the radius of convergence just by looking at the question. Thank-you for your replies Any more hinters as altough I'm trying my best I'm still not getting it . x

6. Originally Posted by mountaindream
No I have not learnt about that kind of series yet.
What do you know about power series?

~It just seems it should be obvious because the question says "state" which implies it should be simple to detect the radius of convergence just by looking at the question. Thank-you for your replies Any more hinters as altough I'm trying my best I'm still not getting it . x
No its not. If you expand a function as a power series about some point, the radius of convergence depends on which point you choose.

Now if you specify that the expansion is about x=0 then you might get an answer.

RonL

7. Originally Posted by CaptainBlack
What do you know about power series?

No its not. If you expand a function as a power series about some point, the radius of convergence depends on which point you choose.

Now if you specify that the expansion is about x=0 then you might get an answer.

RonL
ok for the first expansion I got: 1+6x+6x^2-4x^3
for 2.I got: -root6+1/root12x-1/root48x^2

Could you tell me where to go from here to state the radius of convergence? I'm struggling with the concept as I dont understand it. I only have a text book explanatin which is very brief!
Responses greatly appreciated
x

8. Originally Posted by mountaindream
No I have not learnt about that kind of series yet. ~It just seems it should be obvious because the question says "state" which implies it should be simple to detect the radius of convergence just by looking at the question. Thank-you for your replies Any more hinters as altough I'm trying my best I'm still not getting it . x
Are you doing binomial expansions?

Then the first is:

$\displaystyle f(x)=1+\left( 3/2 \right) (4x) + \frac{\left( (3/2)(3/2-1) \right)}{2!} (4x)^2+ \frac{\left( (3/2)(3/2-1)(3/2-2) \right)}{3!} (4x)^3+...$

which needs a bit of simplifying

RonL

9. Originally Posted by CaptainBlack
Are you doing binomial expansions?

Then the first is:

$\displaystyle f(x)=1+\left( 3/2 \right) (4x) + \frac{\left( (3/2)(3/2-1) \right)}{2!} (4x)^2+ \frac{\left( (3/2)(3/2-1)(3/2-2) \right)}{3!} (4x)^3+...$

which needs a bit of simplifying

RonL
yes I got that result and then simplified it to get the polynomail whihc I stated previously but I dont understand how to find the radius of convergence from that :S
Thankyou x

10. The radius of convergence of the binomial expansion of $\displaystyle (1+x)^a$, where $\displaystyle a$ is not a non-negative integer is $\displaystyle 1$.

RonL