Math Help - Simple Taylor series problem

1. Simple Taylor series problem

Hi i just got introduced to Taylor and Maclaurin series in my calculus class and it was a very short explanation at the end of class. When i got to do doing my homework i found i had no idea where to begin, can someone help? i got the base idea but what to do is beyond me

Find Taylor series for f(x) centered at the given value of a[assume that f has power series expansion. do not show that r_n_(x) goes to 0]

f(x) = 1 + x + x^2
Centered at a = 2

2. Originally Posted by usvn
Hi i just got introduced to Taylor and Maclaurin series in my calculus class and it was a very short explanation at the end of class. When i got to do doing my homework i found i had no idea where to begin, can someone help? i got the base idea but what to do is beyond me

Find Taylor series for f(x) centered at the given value of a[assume that f has power series expansion. do not show that r_n_(x) goes to 0]

f(x) = 1 + x + x^2
Centered at a = 2
f(2) = 7.

f'(2) = 5.

f''(2) = 2

All higher derivatives are zero.

So your series is $f(x) = f(2) + (x - 2) f'(2) + (x - 2)^2 \frac{f''(2)}{2!} = \, ....$

You can check the final answer by expanding it all out and simplifying - you should get $1 + x + x^2 \, ....$

3. Thanks a lot

4. Knowing that this is a polynomial of degee 2, so that its "Taylor series", about any point, is just another polynomial of degree 2, you could also write u= x- 2 so that x= u+2:

$f(x)= 1+ x+ x^2= 1+ (u+2)+ (u+2)^2= 1+ u+ 2+ u^2+ 4u+ 4$
$= u^2+ 5u+ 7= (x- 2)^2+ 5(x- 2)+ 7$