# Talyor Poly. of Degree 2n

• February 23rd 2010, 02:32 PM
ABigSmile
Talyor Poly. of Degree 2n
$f(x) = cosx$, $n \in \mathbb{N}$

Find the Taylor Poly. of $P_{2n}:= {P_{2n}}^{f,0}$
I know $x_0 = 0$ and $f^{(k)}(x) = cosx$ and $f^{(k)}(0) = cos(0) = 1$ and $P_{2n} = \displaystyle\sum_{k=0}^{2n} \frac{f^{(k)}(x_0)}{k!} \cdot (x-x_0)^k = f(0) + \sum_{k=1}^{2n} \frac{f^{(k)}(0)}{k!} \cdot x^k = 1+ \sum_{k=1}^{2n} \frac{x^k}{k!}$

So I am confused on what I should do from here to continue (if what I did previously was correct) because I know this is not complete but I do not know where to go from here, give me a boost please. Thank you
• February 23rd 2010, 03:41 PM
Drexel28
Quote:

Originally Posted by ABigSmile
$f(x) = cosx$, $n \in \mathbb{N}$

Find the Taylor Poly. of $P_{2n}:= {P_{2n}}^{f,0}$
I know $x_0 = 0$ and $f^{(k)}(x) = cosx$ and $f^{(k)}(0) = cos(0) = 1$ and $P_{2n} = \displaystyle\sum_{k=0}^{2n} \frac{f^{(k)}(x_0)}{k!} \cdot (x-x_0)^k = f(0) + \sum_{k=1}^{2n} \frac{f^{(k)}(0)}{k!} \cdot x^k = 1+ \sum_{k=1}^{2n} \frac{x^k}{k!}$

So I am confused on what I should do from here to continue (if what I did previously was correct) because I know this is not complete but I do not know where to go from here, give me a boost please. Thank you

$f^{k}(0)=1$?
• February 23rd 2010, 06:55 PM
southprkfan1
Quote:

Originally Posted by ABigSmile
$f(x) = cosx$, $n \in \mathbb{N}$

Find the Taylor Poly. of $P_{2n}:= {P_{2n}}^{f,0}$
I know $x_0 = 0$ and $f^{(k)}(x) = cosx$ and $f^{(k)}(0) = cos(0) = 1$ and $P_{2n} = \displaystyle\sum_{k=0}^{2n} \frac{f^{(k)}(x_0)}{k!} \cdot (x-x_0)^k = f(0) + \sum_{k=1}^{2n} \frac{f^{(k)}(0)}{k!} \cdot x^k = 1+ \sum_{k=1}^{2n} \frac{x^k}{k!}$

So I am confused on what I should do from here to continue (if what I did previously was correct) because I know this is not complete but I do not know where to go from here, give me a boost please. Thank you

First, your statement is not correct:

f(x) = cosx --> f(0) = 1
f'(x) = -sinx --> f'(0) = 0
f''(x) = -cosx --> f''(0) = -1
f'''(x) = sinx --> f'''(0) = 0

and repeat