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Problem 736

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

Expert verified
The first four terms of the expansion of $$(2-1)^{\frac{1}{2}}$$ are: $$\frac{1}{\sqrt{2}} - \frac{1}{2} - \frac{1}{8\sqrt{2}} - \frac{5}{32}$$
See the step by step solution

## 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#

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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