Question

Taylor series,


So taylor series are good for approximating functions.


How is it possible to approximate a function using only the derivatives of just one of its points ?


Say that I have a function A and B where the have they same slope and value at point x_1 .

now lets take infinite taylor terms of A and B .

what is that taylor polynomial gonna look like? Is it gonna look like A? or B?

Answer 1

well, two things
1. Taylor series don't always converge.
2. You're looking at not just the first derivative, but the NTH derivative. So if you know the nth derivative of the function...

Answer 2

but the nth derivate of the same point

Answer 3

yes, which is why taylor series don't always converge, though they often do.

Answer 4

so if they dont converge , then I may get A at one point and B at another point in time

Answer 5

Well, you won't necessarily get B, you'll probalby get some C...

Answer 6

for instance, you can't make a taylor series converge for 1/x last I checked.

Answer 7

by converging you mean oscilating right

Answer 8

by converging, I mean the taylor series does not equal the function itself.
check https://en.wikipedia.org/wiki/Taylor_series#Approximation_and_convergence

Answer 9

is it possible that the 60th degree polinomial will approximate the function better than the 61th ?

Answer 10

Yes. depends on the radius of convergence

Answer 11

i see