Chapter 1: Q12P (page 36)

Show as in Problem 11that the Maclaurin series for $e^{x}$ converges to $e^{x}$ .

Short Answer

Expert verified

The Maclaurin series for $e^{x}$ converges to $e^{x}$ .

Step by step solution

Given Information

The function is $e^{x}$ .

Definition of the convergence of the Series

The Series is said to be convergent if the terms of the Series are moving toward zero.

Verify the statement

To prove the Maclaurin series for $e^{x}$ converges to $e^{x}$ , we need to show that

$f (X) = \lim_{n \to \infty} T_{n} (X)$ which is equivalent to showing that the remainder

$R_{n} (X) = f (X) - T_{n} (X) \to 0$ . The function is $f (X) = e^{x}$ .

Plugging this into the inequality, and taking the limit, we have:

$\lim_{n \to \infty} |R_{n} (X)| \leq \lim_{n \to \infty} \frac{x^{n + 1} e^{x}}{(n + 1)!} = e^{x} \lim_{n \to \infty} \frac{x^{n + 1}}{(n + 1)!} = 0$

Hence, $\lim_{n \to \infty} \frac{x^{n + 1} e^{x}}{(n + 1)!} = 0 and T_{n} (X) \to f (X) .$ and .

Thus, the Maclaurin series for $e^{x}$ converges to $e^{x}$ .

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Show as in Problem 11that the Maclaurin series for $e^{x}$ converges to $e^{x}$ .

Short Answer

Step by step solution

Given Information

Definition of the convergence of the Series

Verify the statement

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Show as in Problem 11that the Maclaurin series forexconverges toex.

Short Answer

Step by step solution

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Definition of the convergence of the Series

Verify the statement

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Most popular questions from this chapter

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Show as in Problem 11that the Maclaurin series for $e^{x}$ converges to $e^{x}$ .