Chapter 1: Q34p (page 32)

In $(2 - e^{- x})$

Short Answer

Expert verified

$\ln (1 + [1 - e^{- x}]) = x - x^{2} + x^{3} - \frac{13 x^{4}}{12} + \dots$

Step by step solution

Given information.

The Maclaurin expansion of the function $\ln (2 - e^{- x})$ is to be evaluated.

Definition of Maclaurin series.

The definition of the Maclaurin series is defined.

$1 - e^{x} = x - \frac{x^{2}}{2} + \frac{x^{3}}{6} - \frac{x^{4}}{24} \dots$

$l n (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots$

Part-A Step 3: Begin by stating the Maclaurin series.

State the Maclaurin series.

$\ln (2 - e^{- x}) = \ln (1 + [1 - e^{- x}])$

$e^{- x} = 1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{6} + \frac{x^{4}}{24} \dots$

$1 - e^{x} = x - \frac{x^{2}}{2} + \frac{x^{3}}{6} - \frac{x^{4}}{24} \dots$

$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots$

Part-B Step 3: Use the Maclaurin series to evaluate.

Replace $x$ with $[x - \frac{x^{2}}{2} + \frac{x^{3}}{6} - \frac{x^{4}}{24} \dots]$ and evaluate the Maclaurin series.

$n (1 + [1 - e^{- x}]) = [x - \frac{x^{2}}{2} + \frac{x^{3}}{6} - \frac{x^{4}}{24} \dots] - \frac{{[x - \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots]}^{2}}{2} + \frac{{[x - \frac{x^{2}}{2} + \dots]}^{3}}{3} - \frac{[x + \dots]^{4}}{4} + \dots$

$\ln (1 + [1 - e^{- x}]) = \frac{(36 x - 36 x^{2} + 36 x^{3} - 39 x^{4})}{36} + \dots$

$\ln (1 + [1 - e^{- x}]) = x - x^{2} + x^{3} - \frac{13 x^{4}}{12} + \dots$

Thus, the final answer is $\ln (1 + [1 - e^{- x}]) = x - x^{2} + x^{3} - \frac{13 x^{4}}{12} + \dots$

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In $(2 - e^{- x})$

Short Answer

Step by step solution

Given information.

Definition of Maclaurin series.

Part-A Step 3: Begin by stating the Maclaurin series.

Part-B Step 3: Use the Maclaurin series to evaluate.

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In(2-e-x)

Short Answer

Step by step solution

Given information.

Definition of Maclaurin series.

Part-A Step 3: Begin by stating the Maclaurin series.

Part-B Step 3: Use the Maclaurin series to evaluate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Rotational Dynamics

Electric Charge Field and Potential

Magnetism and Electromagnetic Induction

Fundamentals of Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

In $(2 - e^{- x})$