Chapter 1: Q-16-24MP (page 1)

Use the series you know to show that:
$l n 3 + \frac{{(l n 3)}^{2}}{2!} + \frac{{(l n 3)}^{3}}{3!} + \dots = 2$

Short Answer

Expert verified

It is shown that the expansion of series $\ln (3) + \frac{\ln {(3)}^{2}}{2!} + \frac{\ln {(3)}^{3}}{3!} + \frac{\ln {(3)}^{4}}{4!} + \dots is 2$

Step by step solution

Maclaurin series and the given series.

Series is $\ln (3) + \frac{\ln {(3)}^{2}}{2!} + \frac{\ln {(3)}^{3}}{3!} + \frac{\ln {(3)}^{4}}{4!} + \dots$
The Maclaurin series ofis given below:

$\begin{array}{rcl} e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots e^{x} - 1 = x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots \end{array}$

Use the Maclaurin series.

Substitute $\ln (3)$ for $x$ in the Maclaurin series of $e^{x} - 1 = x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots$ as follows:

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

The value of $e^{\ln (3)}$ is given below:

$e^{\ln (3)} = 3$

Substitute $3$ for $e^{\ln (3)}$ in the equation $e^{\ln (3)} - 1 = \ln (3) + \frac{\ln {(3)}^{2}}{2!} + \frac{\ln {(3)}^{3}}{3!} + \frac{\ln {(3)}^{4}}{4!} + \dots$ as follows:

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

Thus, it is shown that the expansion of series $\ln (3) + \frac{\ln {(3)}^{2}}{2!} + \frac{\ln {(3)}^{3}}{3!} + \frac{\ln {(3)}^{4}}{4!} + \dots is 2$

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Use the series you know to show that:
$l n 3 + \frac{{(l n 3)}^{2}}{2!} + \frac{{(l n 3)}^{3}}{3!} + \dots = 2$

Short Answer

Step by step solution

Maclaurin series and the given series.

Use the Maclaurin series.

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Use the series you know to show that:ln3+(ln3)22!+(ln3)33!+…=2

Short Answer

Step by step solution

Maclaurin series and the given series.

Use the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

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Use the series you know to show that:
$l n 3 + \frac{{(l n 3)}^{2}}{2!} + \frac{{(l n 3)}^{3}}{3!} + \dots = 2$