Proving Continuity & Finding MacLorin Polynom of f(x)

[TheForumLord]

Homework Statement

A. Let $$g(x)= \sigma\frac{1}{sqrt(n)}(x^{2n}-x^{2n+1})$$
Prove g(x) is continuous in [0,1].

B. Let f be a function such as f(0)=1 and there's a neighouhood of x=0 in which :
$$f ' (x)= 1+(f(x))^{10}$$ .
Find the MacLorin Polynom of degree 3 of f(x).

Homework Equations

The Attempt at a Solution

Have no idea about it...

Thanks in advance

 

[Mark44]

For A you need to show that g is continuous at each point in [0, 1]. What's the definition of continuity of a function at a point? Do you have to do this by using the definition of continuity or can you use the fact that this is a polynomial and all polynomials are everywhere continuous?

For B you need f(0), f'(0), f''(0), and f'''(0). You already are given that f(0) = 1, and you have f'(x), which you can evaluate at x = 0.

To find f''(0), you need to find f''(x), which you can do by differentiating f'(x), and then evaluate f''(x) at x = 0.
To find f'''(0), differentiate f''(x), and then evaluate at x = 0.

 

[LCKurtz]

[b

B. Let f be a function such as f(0)=1 and there's a neighouhood of x=0 in which :
$$f ' (x)= 1+(f(x))^{10}$$ .
Find the MacLorin Polynom of degree 3 of f(x).

That's "Maclaurin Polynomial".

Well, don't you just need f(0), f'(0), f''(0) and f''(0) to calculate that series? You are given formulas for f(0) and f'(x). You must need a couple more derivatives. You might need the chain rule. Show us your derivatives.

 

[TheForumLord]

Well, B is completely understandable...
About A->I need to show it's continuous using power-series theorems...If I'll prove that the given power-series is convergeing uniformly to g - I'll be done...I've no idea about it...
I'll be delighted to get some help

Thanks!

 

[HallsofIvy]

Power series? g(x) is a polynomial! You don't need to worry about any power series or convergence! A is not asking about a limit as n goes to infinity is it? It is just about a single polynomial for a fixed value of n.

Or was that $\sigma$ supposed to be $\Sigma$? That is, is this a sum over all n? In that case, because it is a power series, it converges uniformly inside its radius of convergence. You need only show that its radius of convergence includes [0, 1].

 

[TheForumLord]

That's excatly what I can't understand...how can I find the eadius of con. in this specific case?


tnx

 

[Mark44]

TheForumLord,
A fair amount of time has been wasted because we didn't understand what you were trying to communicate. It has now come to light that your first problem problem is a summation. The Greek alphabet has upper case letters and lower case letters. In particular, upper case sigma, $\Sigma$, is used to represent summations. Lower case sigma, $\sigma$, is used in statistics to represent the population standard deviation. I interpreted $\sigma$ in this problem as a constant. It didn't occur to me that you really meant a summation.

Also, at this stage of your mathematical education, you really ought to learn how to spell "calculus." It's clear to me that you're not likely to be in the finals of a math spelling bee, but at least get calculus right.

 

[TheForumLord]

Dear Mark44... My english is pretty lame indeed but in this particular case, writing calculus in a wrong way was just a typing mistake - which can occur to anyone...

I didn't know how to write Upper case sigma in Latex so please don't judge me...

 

[Mark44]

Dear Mark44... My english is pretty lame indeed but in this particular case, writing calculus in a wrong way was just a typing mistake - which can occur to anyone...

Sure, anyone can make a typo, but you can eliminate at least some of them by looking over what you've written before you hit the submit button.

I didn't know how to write Upper case sigma in Latex so please don't judge me...

[ tex] \sigma[/tex] or [ itex] \sigma[/itex] (without the leading space) produces $\sigma$.
[ tex] \Sigma[/tex] or [ itex] \Sigma[/itex] (without the leading space) produces $\Sigma$.

Same for all the rest of the Greek letters.

 

[TheForumLord]

Thanks

 

[HallsofIvy]

Better is \sum :
$$\sum$$.

By the way, just clicking on a formula in any post will show you the LaTex code used for it.

Your series can be written by separating the even and odd powers- it is $\sum a_mx^m$ with
$$a_m= \sqrt{\frac{2}{m}}$$
if m is even and
$$a_m= \sqrt{\frac{2}{m-1}}$$
if m is odd.
As for finding the radius of convergence, using the ratio test gives
$$|x|< \sqrt{\frac{m+1}{m}}$$
if n= 2m and
$$|x|< \sqrt{\frac{m}{m-1}}$$
if n= 2m+1

What is the limit of those as n goes to infinity?

Of course, you will need to check if the sum converges at x= 1 but that is easy.