6.3 Taylor and maclaurin series  (Page 3/13)

Taylor’s theorem with remainder

Recall that the n th Taylor polynomial for a function $f$ at a is the n th partial sum of the Taylor series for $f$ at a . Therefore, to determine if the Taylor series converges, we need to determine whether the sequence of Taylor polynomials $\left\{p_n\right\}$ converges. However, not only do we want to know if the sequence of Taylor polynomials converges, we want to know if it converges to $f.$ To answer this question, we define the remainder $R_n\left(x\right)$ as

For the sequence of Taylor polynomials to converge to $f,$ we need the remainder R _n to converge to zero. To determine if R _n converges to zero, we introduce Taylor’s theorem with remainder    . Not only is this theorem useful in proving that a Taylor series converges to its related function, but it will also allow us to quantify how well the n th Taylor polynomial approximates the function.

Here we look for a bound on $\left|R_n\right|.$ Consider the simplest case: $n=0.$ Let p ₀ be the 0th Taylor polynomial at a for a function $f.$ The remainder R ₀ satisfies

If $f$ is differentiable on an interval I containing a and x , then by the Mean Value Theorem there exists a real number c between a and x such that $f\left(x\right)-f\left(a\right)=f^{\prime }\left(c\right)\left(x-a\right).$ Therefore,

Using the Mean Value Theorem in a similar argument, we can show that if $f$ is n times differentiable on an interval I containing a and x , then the n th remainder R _n satisfies

for some real number c between a and x . It is important to note that the value c in the numerator above is not the center a , but rather an unknown value c between a and x . This formula allows us to get a bound on the remainder R _n . If we happen to know that $\left|f^{\left(n+1\right)}\left(x\right)\right|$ is bounded by some real number M on this interval I , then

for all x in the interval I .

We now state Taylor’s theorem, which provides the formal relationship between a function $f$ and its n th degree Taylor polynomial $p_n\left(x\right).$ This theorem allows us to bound the error when using a Taylor polynomial to approximate a function value, and will be important in proving that a Taylor series for $f$ converges to $f.$

Proof

Fix a point $x\in I$ and introduce the function g such that

We claim that g satisfies the criteria of Rolle’s theorem. Since g is a polynomial function (in t ), it is a differentiable function. Also, g is zero at $t=a$ and $t=x$ because