1. ## Taylor's development question

Is there a way to obtain the formula of the Taylor's expression of continuous functions. For example, how can I obtain the Taylor's development of the function $\sqrt{x}$ without trying to find it by writing down the first derivative, the second one, etc?

2. ## WEll

Originally Posted by arbolis
Is there a way to obtain the formula of the Taylor's expression of continuous functions. For example, how can I obtain the Taylor's development of the function $\sqrt{x}$ without trying to find it by writing down the first derivative, the second one, etc?
What you should do is memorize all the Maclaurin formulae then you can go.."what is $e^{x}$" and bam you know its $\sum_{n=0}^{\infty}\frac{x^n}{n!}$ and then you will go 'what is the power series for $e^{-x^2}$?" and you will go "it is $\sum_{n=0}^{\infty}\frac{(-x^2)^{n}}{n!}=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{n!}$"

3. Ok, thank you. So there's no formal way to obtain these formulas. I must write down the first derivatives and try to guess the nth one. Anyway, I'm having some trouble with the $\sqrt{x}$'s one. I would appreciate if you give me the formula .

4. ## No

Originally Posted by arbolis
Ok, thank you. So there's no formal way to obtain these formulas. I must write down the first derivatives and try to guess the nth one. Anyway, I'm having some trouble with the $\sqrt{x}$'s one. I would appreciate if you give me the formula .
If you read what I said you could just memorize the power series...and imput the different values...and for $\sqrt{x}$ I would suggest but am not positve you would use $(1+{x-1})^{\alpha}$ with $\alpha=\frac}1}{2}$...and then apply this Taylor series - Wikipedia, the free encyclopedia look for the one that looks like the one I gave you with the alpha

5. Originally Posted by arbolis
Ok, thank you. So there's no formal way to obtain these formulas. I must write down the first derivatives and try to guess the nth one. Anyway, I'm having some trouble with the $\sqrt{x}$'s one. I would appreciate if you give me the formula .

I haven't to time to give a detailed response right now, but this should be enough to get you started.

Firstly when you take your first few derivatives do not attempt the simplify the fractions that may lead to you missing the general formula.

also, for this problem you will need to come up with a formula for the product first n odd numbers, can you see why ?

If you need help with that just ask.

Bobak

6. also, for this problem you will need to come up with a formula for the product first n odd numbers, can you see why ?
I don't understand well what you mean. I've wrote some derivatives of $\sqrt{x}$. I'm trying to guess the nth derivative, and till now what I have is $(-1)^{n}(1-2n)x^{1/2-n}/2^n$ but I know I'm missing a product in the numerator.

7. I think I got it. If you replace the (1-2n) in my previous formula by the product of (2i-1), from 0 to n, I think it works! Since I've to go now, I'll check that later.