Help with a HW problem.

[Sorc.Polgara]

Ok, I need help with a HW problem:

Find the series representation of the following function:

f(x) = 5x^3 - 13x^2 + 7

Now, this problem derives from section 8.6 in my textbook, which is about representations of functions as power series, however I am unable to find any examples in the textbook where it shows how to represent such a polynomial function as a power series.

Up until this problem, I've been successful in representing functions such as f(x) = 1/(1+x^4) and f(x) = 1/(1+4x^2) because I could relate the function easily to f(x) = 1/(1-x), which has a known power series of  ∑ x^n.

Looking over the other HW problems I notice that in order for me to do those it's a prerequisite that I understand how to represent such polynomial functions as power series.

I'd greatly appreciate it if someone could explain to me how to represent the quoted problem as a power series.  Thanks in advance.

[Yoni]

Degree 0:
P0(x) = 7

Degree 1:
P1(x) = 7

Degree 2:
P2(x) = 7 - 13x^2

Degree 3 or any degree n, n >= 3:
Pn(x) = 7 - 13x^2 + 5x^3

i.e.,
f(x) = ∑ a_n x^n, where a_0 = 7, a_2 = -13, a_3 = 5 and a_k for any other k is 0.

Get it?

[Sorc.Polgara]

Degree 0:
P0(x) = 7

Degree 1:
P1(x) = 7

Degree 2:
P2(x) = 7 - 13x^2

Degree 3 or any degree n, n >= 3:
Pn(x) = 7 - 13x^2 + 5x^3

i.e.,
f(x) = ∑ a_n x^n, where a_0 = 7, a_2 = -13, a_3 = 5 and a_k for any other k is 0.

Get it?

So what you're saying is that I'm suppose to find a pattern between the a_n values, which are the coefficients, and use this pattern to find a function to represent all values of a_n?

I've been trying to do this for the last two hours via trial and error but to no avail... my eraser is kind of non-existent atm too.  I can't seem to find a consistent pattern for all values of a_n...

Sigh.  This is frustrating me.

[Rule]

You're given the power series in the question!  Carefully think about what a power series is and then re-read Yoni's post.

The power series representation of 7 - 13x^2 + 5x^3 is 7 - 13x^2 + 5x^3; a power series does not have to be expressable in the form a*1/[1-f(x)]

Edit:
(For c = 0 (see Wikipedia link)),
P(x) = a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴ + ... + a_nxⁿ
(Note that this is a series so there are supposedly an infinite number of terms, even if there are only finitely many non-zero terms).

So matching 7 - 13x^2 + 5x^3 with the above expression,
a₀ = 7
a₁ = 0
a₂ = 13
a₃ = 5
For k>=4, a_k = 0

[Sorc.Polgara]

You're given the power series in the question!  Carefully think about what a power series is and then re-read Yoni's post.

The power series representation of 7 - 13x^2 + 5x^3 is 7 - 13x^2 + 5x^3; a power series does not have to be expressable in the form a*1/[1-f(x)]

Edit:
(For c = 0 (see Wikipedia link)),
P(x) = a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴ + ... + a_nxⁿ
(Note that this is a series so there are supposedly an infinite number of terms, even if there are only finitely many non-zero terms).

So matching 7 - 13x^2 + 5x^3 with the above expression,
a₀ = 7
a₁ = 0
a₂ = 13
a₃ = 5
For k>=4, a_k = 0

Doh, I feel like a retard now.

Thank you very much Yoni, as well as you Rule for the clarification.

The reason why I probably got confused in the first place was that there is a second part to the question which asks you to find the radius of convergence for it.  This is because, again, based on the examples in the textbook and examples done in class/notes, I couldn't figure out how to find the radius of convergence for it due to the fact that the examples were all of power series sum form (i.e. ∑ x^n) and the radius of convergence was then found using the Ratio Test.  Thus I got confused because I couldn't find a possible way of using the Ratio Test on the power series 7 - 13x^2 + 5x^3.

Anyways, the graded homework has already been turned in and I haven't received it back yet.  I was crammed on time so I made a quick guess and thought that a possible answer could be that it didn't have a radius of convergence.  Oh well.

Again thank you both, I appreciate all of your efforts in helping a newb like me.

[Rule]

You're given the power series in the question!  Carefully think about what a power series is and then re-read Yoni's post.

The power series representation of 7 - 13x^2 + 5x^3 is 7 - 13x^2 + 5x^3; a power series does not have to be expressable in the form a*1/[1-f(x)]

Edit:
(For c = 0 (see Wikipedia link)),
P(x) = a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴ + ... + a_nxⁿ
(Note that this is a series so there are supposedly an infinite number of terms, even if there are only finitely many non-zero terms).

So matching 7 - 13x^2 + 5x^3 with the above expression,
a₀ = 7
a₁ = 0
a₂ = 13
a₃ = 5
For k>=4, a_k = 0

Doh, I feel like a retard now.

Thank you very much Yoni, as well as you Rule for the clarification.

The reason why I probably got confused in the first place was that there is a second part to the question which asks you to find the radius of convergence for it.  This is because, again, based on the examples in the textbook and examples done in class/notes, I couldn't figure out how to find the radius of convergence for it due to the fact that the examples were all of power series sum form (i.e. ∑ x^n) and the radius of convergence was then found using the Ratio Test.  Thus I got confused because I couldn't find a possible way of using the Ratio Test on the power series 7 - 13x^2 + 5x^3.

Anyways, the graded homework has already been turned in and I haven't received it back yet.  I was crammed on time so I made a quick guess and thought that a possible answer could be that it didn't have a radius of convergence.  Oh well.

Again thank you both, I appreciate all of your efforts in helping a newb like me.

"Interval of convergence" would probably be more appropriate for the series you are considering ('radius' is often used to describe a ball of convergence in the complex plane for complex valued series). 

In general, your problems seem to stem from clinging too closely to special cases
or specific examples.    So, think about what it means to converge in general rather than all of the tests that people use.

For example, when is x+ x^2 + x^3 + x^4 + x^5. + ... + x^n a finite number?
Clearly when |x| is less than 1.  Then you could show this a little more rigorously using the ratio test.

When is 7 - 13x^2 + 5x^3 finite?  When x is finite.  So the interval (or radius) of convergence is infinite (the series converges for all x in the set (-infinity, infinity)).

[Yoni]

This was a great question for demonstrating a real understanding of power series rather than memorizing and manipulating specific formulas. Good for us!