Home > Calculus > Lagrange Form of the Remainder Term in a Taylor Series

How do you find the Remainder term in Taylor Series?

Answer 1

There are lots of different ways this can be thought about, though probably the simplest and most common way is to use the inequality #|R_{k}(x)|\leq \frac{M\cdot r^{k+1}}{(k+1)!}#, where the #(k+1)#st derivative #f^{(k+1)}(x)# of #f# satisfies #|f^{(k+1)}(x)|\leq M# over the interval #[c-r,c+r]# (assuming sufficient differentiability/smoothness of #f# over the interval).

The function #R_{k}(x)# is the "remainder term" and is defined to be #R_{k}(x)=f(x)-P_{k}(x)#, where #P_{k}(x)# is the #k#th degree Taylor polynomial of #f# centered at #x=a#: #P_{k}(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots#.

For example, if #f(x)=e^(x)#, #a=0#, and #k=4#, we get #P_{4}(x)=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}#. Moreover, if we consider this over the interval #[-2,2]# so that #r=2#, then the inequality above becomes #|R_{4}(x)|\leq \frac{e^{2}\cdot 2^{5}}{5!}=\frac{4}{15}e^{2}\approx 1.97#. So we can expect #P_{4}(x)# to approximate #e^(x)# to within this amount over the interval #[-2,2]# (actually, the approximation is even better than the error bound gives...the point is that the error bound gives a guarantee ).

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Answer 2

Emilia Aguilar

The remainder term in Taylor series, also known as the remainder or the remainder function, is an expression that represents the difference between a function and its Taylor polynomial approximation. It quantifies the error introduced by using a finite number of terms in the Taylor series to approximate the function.

The most common form of the remainder term in Taylor series is given by the Lagrange form of the remainder, which is expressed using the function's higher-order derivatives.

For a function $f(x)$ that is $(n+1)$ -times differentiable on an interval containing $a$ and $x$ , the Lagrange form of the remainder term $R_n(x)$ in the Taylor series centered at $a$ is given by:

$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}$

where $c$ is some point between $a$ and $x$ .

The remainder term allows us to estimate the error when approximating a function with its Taylor polynomial. By choosing an appropriate number of terms $n$ and analyzing the behavior of the remainder, we can ensure that the Taylor series approximation provides a sufficiently accurate representation of the function within a given interval.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.