# Maclaurin expansion and error

my professor told me any n times differentiable function can be approximated by macularine/taylor expansion.

is that true? As far as I know, if the function is approximated at point a, the approximation is valid if we pick a point near a.

however, if we assume that we picked a point of a+infinite=b, how do we know the approximation can we conclude the approximation is valid or not?

For example, if we approximate a function with macularine series, and picked the point b=infinite for x, shouldn't we check the ratio test first?

ex.lima,n→∞(n+1)(a)*(x-a)(n+1)/(n+1)!]/[ƒ(n)(a)*(x-a)(n)/(n)!]

simplyfying this equation gives

lima,n→∞n+1(a)*(xn+1+ .... +an+1)]/[ƒn(a)*(n+1)*(xn....+an]

if it converges for every continuous smooth n time differetiable function, can anyone provie me proof?[/SUB]

Last edited:

Mark44
Mentor
my professor told me any n times differentiable function can be approximated by macularine/taylor expansion.
That's Maclaurin...
kidsasd987 said:
is that true? As far as I know, if the function is approximated at point a, the approximation is valid if we pick a point near a.

however, if we assume that we picked a point of a+infinite=b
A Maclaurin series or a Taylor series has an interval of convergence, centered at a. Some series converge only at a, and others converge on an interval (a - r, a + r). Still others converge over the entire real number line.
kidsasd987 said:
, how do we know the approximation can we conclude the approximation is valid or not?
If we're evaluating the series at a point within its interval of convergence.
kidsasd987 said:
For example, if we approximate a function with macularine series, and picked the point b=infinite for x, shouldn't we check the ratio test first?

ex.lima,n→∞(n+1)(a)*(x-a)(n+1)/(n+1)!]/[ƒ(n)(a)*(x-a)(n)/(n)!]

if it converges for every continuous smooth n time differetiable function, can anyone provie me proof?

That's Maclaurin...
A Maclaurin series or a Taylor series has an interval of convergence, centered at a. Some series converge only at a, and others converge on an interval (a - r, a + r). Still others converge over the entire real number line.
If we're evaluating the series at a point within its interval of convergence.

yeah.. that was weird cause I found macularine converges but taylor somewhat behaves differently as x,a,n three variables reach to infinity

Mark44
Mentor
yeah.. that was weird cause I found macularine converges but taylor somewhat behaves differently as x,a,n three variables reach to infinity
A Maclaurin series is a Taylor series. A Taylor series is an infinite polynomial in powers of x - a. A Maclaurin series is an infinite polynomial in powers of x - 0. There is no such thing as a "macularine" series.
taylor somewhat behaves differently as x,a,n three variables reach to infinity
I have no idea what you're trying to say here. In a Taylor series, a is fixed. If x is outside the interval of convergence then of course the series will fail to converge. The same is true for a Maclaurin series.

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