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The binomial series practice problems

The first terms of the binomial series (1 + u)ʳ for a fractional or negative r. 55 problems with worked solutions; in 55 of them every equation is proved by a computer algebra system.

4 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 7.585

medium

Find the first five terms of the binomial series for \( \displaystyle \sqrt{\frac{x}{2} + 1} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/2 and u = x/2.
Reviewed
\[ \frac{\left(1 \cdot \frac{1}{2}\right)^{2} \left(-1 + 1 \cdot \frac{1}{2}\right)}{2 \cdot 2} = - \frac{1}{32} \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{\frac{5 x^{4}}{2048} - \frac{x^{3}}{128} + \frac{x^{2}}{32} - \frac{x}{4} + \sqrt{\frac{x}{2} + 1} - 1}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( - \frac{5 x^{4}}{2048} + \frac{x^{3}}{128} - \frac{x^{2}}{32} + \frac{x}{4} + 1 + \cdots \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 7.586

medium

Find the first five terms of the binomial series for \( \displaystyle \left(\frac{x}{2} + 1\right)^{\frac{2}{3}} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 2/3 and u = x/2.
Reviewed
\[ \left(1 \cdot \frac{1}{2}\right)^{2} \cdot 2 \left(-1 + \frac{2}{3}\right) \frac{1}{3} \cdot \frac{1}{2} = - \frac{1}{36} \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{\frac{7 x^{4}}{3888} - \frac{x^{3}}{162} + \frac{x^{2}}{36} - \frac{x}{3} + \left(\frac{x}{2} + 1\right)^{\frac{2}{3}} - 1}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( - \frac{7 x^{4}}{3888} + \frac{x^{3}}{162} - \frac{x^{2}}{36} + \frac{x}{3} + 1 + \cdots \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 7.666

medium

Find the first five terms of the binomial series for \( \displaystyle \sqrt{1 - x} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/2 and u = -x.
Reviewed
\[ \frac{\left(-1\right)^{2} \left(-1 + 1 \cdot \frac{1}{2}\right)}{2 \cdot 2} = - \frac{1}{8} \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{\frac{5 x^{4}}{128} + \frac{x^{3}}{16} + \frac{x^{2}}{8} + \frac{x}{2} + \sqrt{1 - x} - 1}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( - \frac{5 x^{4}}{128} - \frac{x^{3}}{16} - \frac{x^{2}}{8} - \frac{x}{2} + 1 + \cdots \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 7.734

medium

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{1 - x}} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/2 and u = -x.
Reviewed
\[ \frac{\left(-1\right)^{2} \left(1 \cdot \frac{1}{2} + 1\right)}{2 \cdot 2} = \frac{3}{8} \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{- \frac{35 x^{4}}{128} - \frac{5 x^{3}}{16} - \frac{3 x^{2}}{8} - \frac{x}{2} - 1 + \frac{1}{\sqrt{1 - x}}}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( \frac{35 x^{4}}{128} + \frac{5 x^{3}}{16} + \frac{3 x^{2}}{8} + \frac{x}{2} + 1 + \cdots \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More the binomial series practice

Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{\frac{x}{2} + 1} \).

Problem 7.584medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - x\right)^{3}} \).

Problem 7.587medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - x\right)^{2}} \).

Problem 7.588medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(x + 1\right)^{2}} \).

Problem 7.589medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(4 x + 1\right)^{\frac{2}{3}} \).

Problem 7.590medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(2 x + 1\right)^{\frac{3}{2}} \).

Problem 7.591medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{1 - 2 x} \).

Problem 7.592medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(x + 1\right)^{\frac{2}{3}} \).

Problem 7.593medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(4 x + 1\right)^{3}} \).

Problem 7.594medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{3 x + 1}} \).

Problem 7.595medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{\frac{x}{2} + 1}} \).

Problem 7.596medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{1 - 2 x}} \).

Problem 7.597medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(1 - x\right)^{\frac{2}{3}} \).

Problem 7.598medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(4 x + 1\right)^{\frac{3}{2}} \).

Problem 7.629medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(1 - 2 x\right)^{\frac{3}{2}} \).

Problem 7.630medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(2 x + 1\right)^{\frac{2}{3}} \).

Problem 7.631medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(\frac{x}{2} + 1\right)^{3}} \).

Problem 7.632medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{x + 1} \).

Problem 7.633medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(3 x + 1\right)^{\frac{2}{3}} \).

Problem 7.634medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(3 x + 1\right)^{\frac{3}{2}} \).

Problem 7.635medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(x + 1\right)^{3}} \).

Problem 7.636medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(\frac{x}{2} + 1\right)^{2}} \).

Problem 7.637medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - 2 x\right)^{3}} \).

Problem 7.638medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{\frac{x}{2} + 1}} \).

Problem 7.654medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(4 x + 1\right)^{2}} \).

Problem 7.655medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt{2 x + 1} \).

Problem 7.656medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{2 x + 1}} \).

Problem 7.657medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(3 x + 1\right)^{2}} \).

Problem 7.658medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(2 x + 1\right)^{3}} \).

Problem 7.659medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{3 x + 1}} \).

Problem 7.660medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt{1 - 2 x} \).

Problem 7.661medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt{x + 1} \).

Problem 7.662medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{4 x + 1}} \).

Problem 7.663medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{1 - 2 x}} \).

Problem 7.664medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt{4 x + 1} \).

Problem 7.665medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(3 x + 1\right)^{3}} \).

Problem 7.667medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{x + 1}} \).

Problem 7.668medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{x + 1}} \).

Problem 7.710medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{4 x + 1} \).

Problem 7.711medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{4 x + 1}} \).

Problem 7.712medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(x + 1\right)^{\frac{3}{2}} \).

Problem 7.713medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - 2 x\right)^{2}} \).

Problem 7.714medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt{3 x + 1} \).

Problem 7.715medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(\frac{x}{2} + 1\right)^{\frac{3}{2}} \).

Problem 7.716medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{3 x + 1} \).

Problem 7.717medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(2 x + 1\right)^{2}} \).

Problem 7.718medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{2 x + 1} \).

Problem 7.719medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \left(1 - 2 x\right)^{\frac{2}{3}} \).

Problem 7.731medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{1 - x} \).

Problem 7.732medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{2 x + 1}} \).

Problem 7.733medium✓ Every equation proved

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{1 - x}} \).

Problem 7.735medium✓ Every equation proved