Question

What is Lagrange Error and how do you find the value for `M`?

I know the formula
`R_n le frac{M}{(n+1)!}|x-a|^(n+1)`, but I'm confused as to how to find `M`.

Answer 1

`M` is the maximum value of `abs(f^((n+1))(xi))` for `xi` in the interval delimited by `x` and `a`.

Explanation:

Consider the Taylor series of a function `f(x)` around `x=a`:

`f(x) = sum_(k=0)^oo f^((k))(a)/(k!) (x-a)^k`

If we stop the Taylor series at `k = n` we have:

`f(x) = P_n(x) +R_n(x)`

where:

`P_n(x) = sum_(k=0)^n f^((k))(a)/(k!) (x-a)^k`

and it can be demonstrated that rest can be expressed as:

`R_n(x) = 1/(n!) int _a^x f^((n+1))(t) (x-t)^n dt`

Applying the second theorem of the mean to this integral we have:

`R_n(x) = 1/((n+1)!) f^((n+1))(xi) (x-a)^(n+1)`

where `xi` is a point between `x` and `a`

Clearly if in the interval delimited by `x` and `a` we have:

`abs(f^((n+1))(xi)) < M`

then:

`abs( R_n(x)) <= M/((n+1)!)abs(x-a)^(n+1)`