Taylor series operator is linear
This article gives a statement of the form that a certain operator from a space of functions to another space of functions is a linear operator, i.e., applying the operator to the sum of two functions gives the sum of the applications to each function, and applying it to a scalar multiple of a function gives the same scalar multiple of its application to the function.
Contents
Statement
Additivity and scalar multiples
- Additivity: Suppose and are functions defined on subsets of reals that are both infinitely differentiable functions at a point which is in the domain of both functions. Then, is infinitely differentiable at and the Taylor series of about is the sum of the Taylor series of at and the Taylor series of at . Note that we add Taylor series formally by adding the coefficients for each power .
- Scalar multiples: Suppose is a function defined on a subset of the reals and it is infinitely differentiable at a point in its domain. Suppose is a real number. Then, is infinitely differentiable at and the Taylor series of the function at is times the Taylor series of at .
Related facts
Similar facts
Facts used
Proof
Proof of additivity
Given: and are functions defined on subsets of reals that are both infinitely differentiable functions at a point which is in the domain of both functions.
To prove: is infinitely differentiable at and the Taylor series of about is the sum of the Taylor series of at and the Taylor series of at .
Proof: By Fact (1), we have:
with equality holding whenever the right side makes sense. Since by assumption, the right side always makes sense, the left side always makes sense, so is infinitely differentiable at and it makes sense to take its Taylor series.
We recall the expressions for the Taylor series:
Thus, the sum of these Taylor series is:
The Taylor series for is:
By , we obtain that these two Taylor series are equal coefficient-wise, hence equal.
Proof of scalar multiples
Given: is a function defined on a subset of the reals and it is infinitely differentiable at a point in its domain. is a real number.
To prove: is infinitely differentiable at and the Taylor series of the function at is times the Taylor series of at .
Proof: By Fact (1), we have that:
with equality holding whenever the right side makes sense. Since by assumption, the right side always makes sense, the left side always makes sense, so is infinitely differentiable and it makes sense to takes its Taylor series.
We note that:
The Taylor series for is:
By , these two Taylor series are equal coefficient-wise, hence equal.