Formal power series & Taylor series

**Last_Singularity** · April 19th 2009, 07:56 AM

I am honestly stuck on these problems - the subjects are neither covered in our textbook nor did I find anything after about an hour of googling:

Question 1: Using manipulations with formal power series, find the Taylor series of the following functions at $x=0$
(a) $\frac{1}{1+x+x^2}$
(b) $sin^{-1}(x)$
(c) $tanh(x)$
(d) $tan(x)$

Question 2: Apply Taylor's formula with Lagrange remainder to estimate the following number with given accuracy (a) cube root of 124 within 0.01 (b) pi within 0.001

Thanks a lot!

**Moo** · April 19th 2009, 08:18 AM

Hello,

Originally Posted by Last_Singularity

I am honestly stuck on these problems - the subjects are neither covered in our textbook nor did I find anything after about an hour of googling:

Question 1: Using manipulations with formal power series, find the Taylor series of the following functions at $x=0$
(a) $\frac{1}{1+x+x^2}$

Well... $1+x+x^2=1-[-x(x+1)]$ and you should know the power series for $\frac{1}{1-x}$

(d) $tan(x)$

Let $\tan(x)=\sum_{n \geq 0} a_n x^n$
Differentiate :
$\frac{1}{1+x^2}=\sum_{n \geq 1} na_n x^{n-1}$
Change the indices to get x^n in the summand :
$\frac{1}{1+x^2}=\sum_{n \geq 0} (n+1)a_{n+1} x^n$
And again, note that $\frac{1}{1+x^2}=\frac{1}{1-(-x^2)}$ and identify the coefficients $a_n$ (or $a_{n+1}$ , that doesn't make much difference)

I really don't know for the other ones... nor am I sure for the methods of these two

I was just giving it a try

**Opalg** · April 19th 2009, 09:01 AM

Originally Posted by Last_Singularity

Question 1: Using manipulations with formal power series, find the Taylor series of the following functions at $x=0$
(a) $\frac{1}{1+x+x^2}$
(b) $sin^{-1}(x)$
(c) $tanh(x)$
(d) $tan(x)$

For (a), I would write it as $\frac{1}{1+x+x^2} = \frac{1-x}{1-x^3} = (1-x)(1-x^3)^{-1}$ . Now use the binomial series $(1-t)^{-1} = 1+t+t^2+t^3+\ldots$ to get $(1-x)(1-x^3)^{-1} = (1-x)(1+x^3+x^6+x^9+\ldots) = 1-x+x^3-x^4+x^6-x^7+\ldots$ .

For (b), notice that $\tfrac d{dx}(\sin^{-1}x) = (1-x^2)^{-1/2}$ . Again use a binomial series, this time for $(1-t)^{-1/2}$ , to get the series for $(1-x^2)^{-1/2}$ . Then integrate it to get the series for $\sin^{-1}x$ .

The series for tan(x) and tanh(x) are both messy. The general term involves Bernoulli numbers (see here). You can get the first few terms by using a method that Moo suggests in the previous comment,namely $\tan x = \frac{\sin x}{\cos x} = (x - \tfrac{x^3}{3!} = \ldots)(1 - (\tfrac{x^2}{2!} - \ldots))^{-1}$ and use the power series for $(1-t)^{-1}$ .

Originally Posted by Last_Singularity

Question 2: Apply Taylor's formula with Lagrange remainder to estimate the following number with given accuracy (a) cube root of 124 within 0.01 (b) pi within 0.001

For the first one, put x=1/125 in the Taylor series for $(1-x)^{1/3}$ .

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