Calculate the Taylor series for (x^5) + (3x^2) - x - 1 about x= 1. Calculate it's radius of convergence. How accurate is the third order approximation on (0,2)? How accurate is the fifth order?

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- Oct 27th 2012, 01:10 PMkandygirl16Taylor series calculation and approximation help please
Calculate the Taylor series for (x^5) + (3x^2) - x - 1 about x= 1. Calculate it's radius of convergence. How accurate is the third order approximation on (0,2)? How accurate is the fifth order?

- Oct 27th 2012, 04:52 PMProve ItRe: Taylor series calculation and approximation help please
A Taylor Series for $\displaystyle \displaystyle \begin{align*} f(x) \end{align*}$ centred around $\displaystyle \displaystyle \begin{align*} x = h \end{align*}$ is given by $\displaystyle \displaystyle \begin{align*} f(x) = \sum_{n = 0}^{\infty} { \frac{ f^{ (n) } (h) }{ n! } \left( x - h \right)^n } \end{align*}$, so calculating the necessary derivatives gives...

$\displaystyle \displaystyle \begin{align*} f(x) &= x^5 + 3x^2 - x - 1 \\ f(1) &= 1^5 + 3 \cdot 1^2 - 1 - 1 &= 2 \\ \\ f'(x) &= 5x^4 + 6x - 1 \\ f'(1) &= 5 \cdot 1^4 + 6\cdot 1 - 1 \\ &= 10 \\ \\ f''(x) &= 20x^3 + 6 \\ f''(1) &= 20\cdot 1^3 + 6 \\ &= 26 \\ \\ f'''(x) &= 60x^2 \\ f'''(1) &= 60 \cdot 1^2 \\ &= 60 \\ \\ f^{\textrm{iv}} (x) &= 120x \\ f^{\textrm{iv}} (1) &= 120 \cdot 1 \\ &= 120 \\ \\ f^{\textrm{v}} (x) &= 120 \\ f^{\textrm{v}} (1) &= 120 \\ \\ f^{\textrm{vi}}(x) &= 0 \\ \vdots \end{align*}$

So your Taylor Series is

$\displaystyle \displaystyle \begin{align*} f(x) &= 2 + 10(x - 1) + \frac{26}{2}(x - 1)^2 + \frac{60}{3!}(x - 1)^3 + \frac{120}{4!}(x - 1)^4 + \frac{120}{5!}(x - 1)^5 \\ &= 2 + 10(x - 1) + 13(x - 1)^2 + 10(x - 1)^3 + 5(x - 1)^4 + (x - 1)^5 \end{align*}$ - Oct 28th 2012, 12:40 PMkandygirl16Re: Taylor series calculation and approximation help please
Thank you very much for your help so far, do you think you could help me with the rest of my question as well????