Chapter 1: Q29p (page 32)

$\sqrt{1 + \ln (1 + x)}$

Short Answer

Expert verified

$\sqrt{1 + \ln (1 + x)} = 1 + \frac{x}{2} - \frac{3 x^{2}}{8} + \frac{17 x^{3}}{48} + \dots$

Step by step solution

Given information

The Maclaurin expansion of the function $\sqrt{1 + \ln (1 + x)}$ is to be evaluated.

Definition of Maclaurin series

The definition of the Maclaurin series is defined.

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

$(1 + x)^{1 / 2} = 1 + \frac{x}{2} - \frac{x^{2}}{8} + \frac{x^{3}}{16} + \dots$

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

State the Maclaurin series.

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

$(1 + x)^{1 / 2} = 1 + \frac{x}{2} - \frac{x^{2}}{8} + \frac{x^{3}}{16} + \dots$

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

Substitute $x$ with $[x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots]$ in the second series and evaluate.

$\sqrt{1 + \ln (1 + x)} = 1 + \frac{[x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots]}{2} - \frac{{[x - \frac{x^{2}}{2} + \dots]}^{2}}{8} + \frac{[x + \dots]^{3}}{16} + \dots$

$\sqrt{1 + \ln (1 + x)} = 1 + \frac{(48 x - 36 x^{2} + 34 x^{3} + \dots)}{96}$

role="math" localid="1657367767479" $\sqrt{1 + \ln (1 + x)} = 1 + \frac{x}{2} - \frac{3 x^{2}}{8} + \frac{17 x^{3}}{48} + \dots$

Thus, the final answer is $\sqrt{1 + \ln (1 + x)} = 1 + \frac{x}{2} - \frac{3 x^{2}}{8} + \frac{17 x^{3}}{48} + \dots$

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$\sqrt{1 + \ln (1 + x)}$

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 first Maclaurin series to evaluate

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1+ln(1+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 first Maclaurin series to evaluate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

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Electricity and Magnetism

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Study anywhere. Anytime. Across all devices.

$\sqrt{1 + \ln (1 + x)}$