Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 15: Representations Of Functions By Power Series – Section 15.6: The Binomial Series


15.6
The Binomial Series

 

 

Return To Contents
Go To Problems & Solutions

 

1. The Binomial Theorem

 

Recall the Binomial Theorem, which is stated as follows and whose proof is omitted.

 

The Binomial Therem

 

 

 

Recall that:

 

 

depending on the calculator. The letter n or k or both may be replaced by other letters, again depending on the calculator.

 

In the binomial expansion, there are n + 1 terms. For the coefficient, n stays fixed, while k increases from 0 to n. As for a, its
exponent decreases from n down to 0. As for b, its exponent increases from 0 up to n. The sum of the exponents of a and b in
any term is always n.

 

Example 1.1

 

Expand (a + b)5 by using the Binomial Theorem.

 

Solution


EOS

 

Remark 1.1

 

If your calculator has a button for the binomial coefficient, you can use it, if the use of the calculator is allowed, instead of
expanding the binomial coefficients into the factorial form.

 

Example 1.2

 

Expand (a – b)5 by utilzing the Binomial Theorem.

 

Solution

EOS

 

Remark 1.2

 

Each term that contains an even power of b is preceded by the “+” sign. This is because –1 raised to an even-number power is
positive 1. The term itself may be either positive or negative, depending on the signs of a and b.

 

Each term that contains an odd power of b is preceded by the “–” sign. This is because –1 raised to an odd-number power is
negative 1. The term itself may be either positive or negative, depending on the signs of a and b.

 

Return To Top Of Page     Go To Problems & Solutions

 

2. The Binomial Series

 

In the binomial expansion of (a + b)n, replacing a by 1 and b by x we get:

 

 

Now after studying Taylor series, we understand that (1 + x)r, where the exponent r is a real number, also has a Maclaurin
series expansion, because clearly (1 + x)r has derivatives of all orders at x = 0. In Theorem 2.1 below, we'll show that the
Maclaurin series of (1 + x)r converges to or represents (1 + x)r for –1 < x < 1, that if r is an integer greater than or equal to 0
then the series is finite (has finitely many non-0 terms), is the same expansion as given by the Binomial Theorem, as shown
above, and represents the function for all x. The series is called the binomial series, of course from the property that it's the
series of the rth power (1 + x)r of the binomial  function 1 + x.

 

In this section we use the letter “k” instead of “n” for the derivative order, since we use the letter “n” for the exponent that's an integer greater than or equal to 0.

 

Theorem 2.1 – The Binomial Series

 

The Maclaurin series of the function (1 + x)r, where r is a real number, represents the function for –1 < x < 1. The
function and the series are:

 

 

 

The proof follows the notes.

 

Notes

1. You should clearly distinguish between these three quantities and shouldn't be confused by them: the variable x, the
    exponent r on 1 + x, and the derivative order k, which is the running index.

 

2. The equation for the series tells that if f(x) = (1 + x)r, then f(0) = 1 and f (k)(0) = r(r – 1)(r – 2)...(r – k + 1) for k = 1, 2,
    3, ... .

 

3. In the product r(r – 1)(r – 2)...(r – k + 1), the factors go down from r to r – (k – 1) = r – k + 1, ie down from r to r minus
    something that's less than k by 1. For k = 1, the product consists of just 1 factor: r to r – 1 + 1 = r, so the product is just
    r. For k = 0, we would have r – k + 1 = r – 0 + 1 = r + 1, and r + 1 is greater than, not less than or equal to, r. So the
    case for k = 0 , for which the term of the series is 1, doesn't apply to the product formula and thus is placed outside the
    sigma summation notation.

 

    The product r(r – 1)(r – 2)...(r – k + 1) consists of k factors, from r (= r – 0) to r – k + 1 (= r – (k – 1), and is in the (k +
    1)st term of the series. The number of factors equals the exponent on x. So as we go from one term to the next, the
    number of factors increases by 1.

 

Proof
Determining The Series
Let f(x) = (1 + x)r. Then:

 

 

Then:

 



Then:

 

 

Then:

 

 

f '(x) + xf '(x) = rf (x),
f '(x)(1 + x) = rf (x),

 

 

 

The series is finite and is the expansion of (1 + x)n as provided by the Binomial Theorem. Thus it “converges” to and represents
(1 + x)n. This representation is for all x, as the Binomial Theorem is valid for all x.

 

EOP

 

Remark 2.1

 

For general real number r, the binomial series:

 

 

Example 2.1

 

 

Solution
 

 

4. The series:

 

 

     A bound on the error size is 0.00086.
EOS

 

Example 2.2

 

Determine the Maclaurin series of arcsin x. Express the series or part of it in the sigma notation. State the interval where the representation applies.

 

Note

 

Solution
 

EOS

 

Return To Top Of Page     Go To Problems & Solutions

 

3. The Binomial Expansion And The Binomial Coefficient

 

We naturally define the binomial expansion of the function (1 + x)r, for any real number r, to be the Maclaurin series of the
function. We will also generalize the definition of the binomial coefficient, which of course is the coefficient r(r – 1)(r – 2)...(r –
k + 1)/k! of the binomial expansion, as shown in Eq. [2.1]. If r is an integer greater than or equal to 0, then the expansion is
finite as given by the Binomial Theorem, and each coefficient can be written in the factorial form r!/(k!(r – k)!), as shown in
the case of (1 + x)n in Eq. [2.3]. Otherwise, no coefficient can, as factorial is defined only for integers that are greater than or
equal to 0. So, to include every real number r, in the definition of the (generalized) binomial coefficient we use the coefficients
r(r – 1)(r – 2)...(r – k + 1)/k! of the series in Eq. [2.1], not the factorial form r!/(k!(r – k)!). Since the product r(r – 1)(r – 2)
...(r – k + 1) excludes the case of k = 0 and the case of k = 0 corresponds to the first term which is 1 = 1x0, we'll distinguish
between the case of k greater than or equal to 1 and the case of k equal to 0.

 

Definition 3.1 – Binomial Expansion And Binomial Coefficient

 

The binomial expansion of (1 + x)r, where r is a real number and –1 < x < 1, is its binomial Maclaurin series that
represents it for –1 < x < 1:

 

 

 

Example 3.1

 

Evaluate the binomial coefficients:

 

 

Solution


EOS

 

Return To Top Of Page

 

Problems & Solutions

 

1. Expand (x + y)6 by using the Binomial Theorem.

 

Solution

 

 

Return To Top Of Page

 

 

2. Expand (x – y)6 by utilizing the Binomial Theorem.

 

Solution

 

 

Return To Top Of Page

 

 

3. Evaluate the following binomial coefficients.

 

 

Solution

 

 

 

Return To Top Of Page

 

 

4. Find the binomial series for (1 + x)0. Verify that the coefficient of xk is the binomial coefficient.

 

Solution

 

 

Return To Top Of Page

 

 

5.

 

Solution

 

 

b. Yes it is, because 7 = 4 + 3 and – 4 < 3 < 4.

 

 

    A bound on the error size is 0.013.

 

Return To Top Of Page

 

 

 

Solution

 

 

Return To Top Of Page

 

 

7. Find the binomial series of:

 

    

 

    Put the series or part of it in the sigma notation. Indicate the interval of valid representation.

 

Solution

 

 

 

Return To Top Of Page     Return To Contents