Solve the problem.
-Use the Alternating Series Estimation Theorem to estimate the error that results from replacing $\mathrm { e } ^ { - \mathrm { x } }$ by $1 - x + \frac { x ^ { 2 } } { 2 }$ when $0

Question

Solve the problem.
-Use the Alternating Series Estimation Theorem to estimate the error that results from replacing $\mathrm { e } ^ { - \mathrm { x } }$ by $1 - x + \frac { x ^ { 2 } } { 2 }$ when $0 < x < 0.2$
A) -0.00133
B) 0.00027
C) 0.00267
D) 0.00133

Quizplus · Accepted Answer

The Alternating Series Estimation Theorem states that the error in approximating a convergent alternating series by its first few terms is no larger than the absolute value of the first neglected term. In this case, the alternating series is

$$\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{n!}.$$

We want to estimate the error when we replace this by the first three terms of its Taylor series, which is

$$1 - x + \frac{x^2}{2}.$$

The first neglected term is

$$\frac{x^3}{3!} = \frac{x^3}{6}.$$

So the error is bounded by

$$\left|\frac{x^3}{6}\right| = \frac{x^3}{6}.$$

The problem asks us to estimate the error when $0 < x < 0.2$, so the maximum possible value of the error is

$$\frac{(0.2)^3}{6} = 0.000133.$$

Since this is positive, the correct answer choice is D) 0.00133.

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