4.2 Binomial expansion for rational n and approximation — Learn

Prerequisites: the binomial expansion for a positive whole number $n$ from Section 4.1; factorials and binomial coefficients; the laws of indices, including fractional and negative exponents from Section 2.1.

By the end of this section you can: expand $(1 + x)^n$ when $n$ is a fraction or a negative number; state the range of $x$ for which the expansion is valid; expand $(a + bx)^n$ by taking out the factor $a^n$; and use the first few terms to approximate a value.

1. The expansion when n is not a positive whole number

The formula from Section 4.1 still works when $n$ is any rational number, but the series never stops. For $r \ge 1$ the coefficient is

The product on the top has $r$ factors. It starts at $n$ and drops by one each time. The series is

and it is valid — it really does add up to $(1 + x)^n$ — only when $|x| < 1$.

The word valid matters. Inside the range, adding more terms brings the sum closer to the true value. Outside it, the series either grows without limit or settles on the wrong number.

2. A fractional power

Example 1. Expand $(1 + x)^{\frac{1}{2}}$ up to and including the term in $x^3$.

Take $n = \frac{1}{2}$ and work out the coefficients one at a time.

So

The signs alternate once the factors become negative. The series continues for ever, so the dots are part of the answer.

3. A negative power

Example 2. Expand $(1 - 2x)^{-1}$ up to and including the term in $x^3$.

Take $n = -1$, so $x$ in the formula is replaced by $-2x$. Every coefficient $\binom{-1}{r} = (-1)^r$.

The condition $|2x| < 1$ is the same as $|x| < \frac{1}{2}$. This series is a geometric series with first term $1$ and ratio $2x$, which is why the terms double each time. Section 4.6 returns to that link.

4. Taking out the factor a to the power n

Most brackets are not in the form $(1 + x)^n$. Write

then expand the bracket. The expansion is valid when

Example 3. Expand $\sqrt{4 + x}$ up to and including the term in $x^3$, and state the values of $x$ for which the expansion is valid.

Take out the factor $4^{\frac{1}{2}} = 2$:

Use the coefficients from Example 1 with $\frac{x}{4}$ in place of $x$:

Work out each term inside the bracket first, then multiply by $2$:

The expansion is valid when $\left|\frac{x}{4}\right| < 1$, which is $|x| < 4$.

5. Using the expansion to approximate

A short expansion gives a good estimate when $x$ is small, because the later powers of $x$ shrink quickly.

Example 4. Use the first three terms of Example 3 to estimate $\sqrt{4.1}$.

The value $4.1$ is $4 + 0.1$, so $x = 0.1$:

A calculator gives $2.0248457$ to $7$ decimal places. The three-term estimate is correct to $5$ decimal places, which shows how quickly the later terms fade.

Example 5. The £2,000 gift in the chapter problem is moved after $2\frac{1}{2}$ years, with the interest still at 6% a year. Estimate its value then.

The value is $2000(1.06)^{2.5}$, and $2.5 = \frac{5}{2}$. Expand $(1 + 0.06)^{\frac{5}{2}}$ with $x = 0.06$:

So

Work out the terms:

Then

so the gift is worth about £2313.63 after $2\frac{1}{2}$ years. The expansion is valid because $|0.06| < 1$.

6. Try it: how far does the approximation reach?

Predict first. The figure draws the true curve of $(1 + x)^n$ and the sum of the first few terms of its expansion. Before you touch the controls, predict what happens to the gap between the two curves as you add more terms, both inside the range $|x| < 1$ and just outside it. Write your prediction down.

Test. Move the slider for the number of terms and watch the dashed curve. Then move the marker to different values of $x$ and read the true value and the approximation in the readout. Choose a negative power and a fractional power from the list and compare them.

Explain. Write two sentences saying why adding terms makes the approximation better inside the range $|x| < 1$ but worse outside it.

Watch out. Two mistakes appear again and again.

Dropping the factor $a^n$. For $\sqrt{4 + x}$ the answer starts at $2$, because $4^{\frac{1}{2}} = 2$. Writing $\left(1 + \frac{x}{4}\right)^{\frac{1}{2}} = 1 + \frac{x}{8} - \dots$ is the expansion of the bracket alone, not of $\sqrt{4 + x}$. Re-check: put $x = 0$. The expansion must give $a^n$, so $\sqrt{4 + 0}$ must give $2$.

Forgetting the range. The expansion of $(1 + x)^n$ is valid only for $|x| < 1$. In the form $(a + bx)^n$ the range is $\left|\frac{bx}{a}\right| < 1$. For $(1 - 2x)^{-1}$ that is $|x| < \frac{1}{2}$, not $|x| < 1$. Re-check: state the range next to your answer, every time.

Progress check

Return to the chapter problem. The committee also considers moving the £2,000 after only 1 year and 6 months, which is $1.5$ years.

1. Write the value as $2000(1.06)^{1.5}$, and write $1.5$ as $\frac{3}{2}$.
2. Use $\binom{\frac{3}{2}}{1}$, $\binom{\frac{3}{2}}{2}$ and $\binom{\frac{3}{2}}{3}$ to expand $(1.06)^{1.5}$ up to the $x^3$ term.
3. Estimate the value after $1.5$ years, to the nearest penny.

You should now be able to

• write the general binomial series for $(1 + x)^n$ with a rational $n$;
• work out $\binom{n}{r}$ for a fractional or negative $n$;
• state the range of $x$ for which an expansion is valid;
• take out the factor $a^n$ from $(a + bx)^n$ and expand the bracket;
• use the first few terms of an expansion to approximate a value.