Chapter 25: Chapter 25

Problem 736

Find the first four terms of the expansion of \((2-1)^{1 / 2}\).

Short Answer

Expert verified

The first four terms of the expansion of \((2-1)^{\frac{1}{2}}\) are: \(\frac{1}{\sqrt{2}} - \frac{1}{2} - \frac{1}{8\sqrt{2}} - \frac{5}{32}\)

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the expression

We have the expression: \((2-1)^{\frac{1}{2}}\) #Step 2: Apply binomial theorem#

Step 2: Apply the binomial theorem

Using the binomial theorem to expand the expression into a series: \((2-1)^{\frac{1}{2}} = \sum_{k=0}^{\infty} \binom{\frac{1}{2}}{k} 2^{\frac{1}{2}-k} (-1)^k \) #Step 3: Find first four terms#

Step 3: Find the first four terms

According to the problem, we only need the first four terms of the expansion. We'll replace k with 0, 1, 2, and 3 successively to find the first four terms: Term 1 (k=0): \(\binom{\frac{1}{2}}{0} 2^{\frac{1}{2}-0} (-1)^0 = \frac{1}{\sqrt{2}}\) Term 2 (k=1): \(\binom{\frac{1}{2}}{1} 2^{\frac{1}{2}-1} (-1)^1 = -\frac{1}{2}\) Term 3 (k=2): \(\binom{\frac{1}{2}}{2} 2^{\frac{1}{2}-2} (-1)^2 = -\frac{1}{8\sqrt{2}}\) Term 4 (k=3): \(\binom{\frac{1}{2}}{3} 2^{\frac{1}{2}-3} (-1)^3 = -\frac{5}{32}\) #Step 4: Write the final answer#

Step 4: Final answer

The first four terms of the expansion of \((2-1)^{\frac{1}{2}}\) are: \(\frac{1}{\sqrt{2}} - \frac{1}{2} - \frac{1}{8\sqrt{2}} - \frac{5}{32}\)

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Chapter 25: Chapter 25

Find the first four terms of the expansion of \((2-1)^{1 / 2}\).

Short Answer

Step by step solution

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Step 1: Write the expression

Step 2: Apply the binomial theorem

Step 3: Find the first four terms

Step 4: Final answer

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