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Chapter 8: Q. 11 (page 692)

What is Taylor’s Theorem?

Short Answer

Expert verified

If f be a function that can be differentiated (n+1) times in some open intervalI that contains the point $x_{0} and also R_{n} (x)$ represents the nth remainder for the function at $x = x_{0}$ .

Step by step solution

01

Step 1. Given Information

The given term is Taylor's polynomial

02

Step 2. Explanation

Taylor's theorem state that if f be a function that can be differentiated (n+1) times in some open intervalI that contains the point $x_{0} and also R_{n} (x)$ represents the nth remainder for the function at $x = x_{0}$ .

Then the nth remainder for function is given by,

$R_{n} (x) = \int_{x_{0}}^{x} \frac{{(x - t)}^{n}}{n!} f^{n + 1} (t) d t$

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Most popular questions from this chapter

If m is a positive integer, how can we find the Maclaurin series for the function $f (c x^{m})$ if we already know the Maclaurin series for the function f(x)? How do you find the interval of convergence for the new series?

Fill in the blanks: The graph of every odd function is symmetric about ______. The graph of every even function is symmetric about ______.

Find the interval of convergence for power series: $\sum_{k = 1}^{\infty} \frac{{(- 1)}^{k}}{k + 1} x^{k}$

Find the interval of convergence for power series: $\sum_{k = 0}^{\infty} \frac{{(- 2)}^{k}}{k!} x^{k}$

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_{4} (x)$

$e^{x}$

See all solutions

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