Question

Can anyone explain to me, a bit on the idea of a remainder of a Taylor Series? I'm using the Purple Book, and Paul's Online Notes, and am just struggling with the whole concept.

Answer 1

do u mean the lagrange rule?

Answer 2

I don't think so, see (CTRL+F "remainder"): http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx

Answer 3

My exact problem, although I'm not just looking for a single solution, more of a concept, is: Let f(x) = e^(-x). Prove that lim n->inf of R (sub) n (x) = 0 and, find a Maclaurin series for f(x). I can do the Maclaurin part, but am having issues visualizing the remainder, R(X).

Answer 4

i have no idea then. sorry!

Answer 5

the remainder is the rest of the stuff you dont use after taking the partial sums you want to take .. right?

Answer 6

I *think*, but I'm not sure. I'm having a hell of a time trying to visualize it.

Answer 7

Here is the wiki on it: http://en.wikipedia.org/wiki/Taylor's_theorem#Explicit_formulae_for_the_remainder

Answer 8

whats the Mac Series youve come up with

Answer 9

or taylor since they are the same except for one says , use a zero, and the other says, use any number

Answer 10

\[1/e + (1/e)x + (1/e2!)x^2 + (1/e3!)x^3 + ...\], Since it's a Maclaurin, it's evaluated at 0. That I get, but I'm not sure what the Remainder is, or how to prove it, as I can't really visualize it.

Answer 11

\[\sum_{n=0}^{inf}\frac{1}{e n!}x^n\]

if i go by your notation i get this as a summation if we just follow the pattern that your series exhibits

Answer 12

now spose we only use up to some finite Nth term:

\[\sum_{n=0}^{inf}\frac{1}{e n!}x^n=\sum_{n=0}^{N}\frac{1}{e n!}x^n+\sum_{n=N+1}^{inf}\frac{1}{e n!}x^n\]

Answer 13

the "remainder" is the stuff that you dont use

Answer 14

the remainder can be calculated and the error made small in this manner; ill use the following to hopefully make a point