Speaker A
Let's take a look at the binomial series. This is one of the most interesting examples of Taylor series. It's the Maclaurin series for the function 1 plus x to the k, where k is some real number. Let's start by looking
Explore the binomial series as a Maclaurin series for (1+x)^k, including coefficients, convergence, and examples for real k values.
Key Takeaways
- The binomial series generalizes the binomial theorem to any real exponent k as an infinite series.
- Binomial coefficients can be computed using factorials or product formulas and relate to combinations.
- The series converges for |x| < 1 and matches known expansions like the geometric series for k = -1.
- Maclaurin series coefficients are derived from successive derivatives evaluated at zero.
- Explicit formulas for coefficients can be complex, but the first few terms are straightforward to calculate.
What the video covers
- Introduction to the binomial series as a Maclaurin series for the function (1+x)^k where k is any real number.
- Explanation of binomial coefficients derived from Pascal's triangle and their combinatorial interpretation as 'k choose i'.
- Derivation of the binomial coefficient formula using factorials and product notation.
- Presentation of the classical binomial theorem for positive integer k as a finite sum.
- Newton's generalization of the binomial theorem to any real k using an infinite series expansion.
- Convergence of the infinite binomial series for |x| < 1 demonstrated via the ratio test.
- Example of k = -1 showing the binomial series reduces to a geometric series with alternating signs.
- Derivation of the Maclaurin series coefficients from the nth derivative of (1+x)^k evaluated at zero.
- Discussion on convergence to the original function and conditions where the series holds.
- Example of k = 1/2 providing the Maclaurin series for the square root function and simplification of coefficients.
Chapters
- 00:00Introduction to the Binomial Series and Positive Integer Powers
- 00:32Pascal's Triangle and Binomial Coefficients
- 01:06Formula for Binomial Coefficients and Examples
- 02:31Classical Binomial Theorem for Positive Integers
- 03:29Generalizing Binomial Coefficients for Real Numbers
- 04:01Infinite Binomial Series and Convergence
- 04:35Example: Binomial Series for k = -1 and Geometric Series
- 05:46Deriving the Maclaurin Series from Derivatives
- 06:54Maclaurin Series Coefficients and Convergence Discussion
- 07:24Example: Binomial Series for k = 1/2 (Square Root Function)
Full Transcript — Download SRT & Markdown
Speaker A
at the case where k is a positive integer. Well, we know 1 plus x squared is 1 plus 2x plus x squared, a familiar result from school. 1 plus x cubed, a little less well-known, is 1 plus 3x
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plus 3x squared plus x cubed. 1 plus x to the fourth is 1 plus 4x plus 6x squared plus 4x cubed plus x to the fourth, and we can carry on multiplying by 1 plus x and getting formulas for
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successive powers of 1 plus x. But a more important question is, where are these coefficients coming from? What are they and how do we calculate them? Well, you may remember that the coefficients come from Pascal's triangle. You see, this last
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row here corresponds to the expansion of 1 plus x to the fourth, and these numbers are obtained by adding up the two numbers above a given number to the left and right. So 3 is 1 plus 2, and 4 is 1
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plus 3, etc. These numbers are called binomial coefficients for obvious reasons, but they also give us the number of ways of choosing a set of i objects out of a set of k objects, so we often call this number k choose i, and the
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formula for it is k factorial divided by k minus i factorial times i factorial. If we write this out, we can cancel out the k minus i factorial with the tail of k factorial, what would have gone here, and
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that just leaves us with k times k minus 1, etc., down to k minus i plus 1. The next term would be k minus i, which is cancelling out down here, and on the bottom we have i factorial, so both of
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these things, the top and the bottom, are products of exactly i numbers. So remember that formula. For instance, 5 choose 2 is 5 factorial divided by 3 factorial times 2 factorial, and then in this form it's 5 times 4 divided by 2 factorial, 2 times 1,
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and we get 10. And that is the coefficient we saw in the expansion of 1 plus x to the fifth. It's the coefficient of x squared. And the classical binomial theorem states that 1 plus x to the power k is the sum from i equals 0 up to
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k of k choose i times x to the i. And if we write that out explicitly, it looks like this: 1 plus kx plus k times k minus 1 over 2 factorial x squared, etc. The coefficients are all the coefficient of x to the i
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will be this number k choose i. And back in the 1600s, Newton observed that this formula could be generalized to any k as long as we extended this to an infinite series rather than just a finite polynomial. So in order to express this
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succinctly, let's generalize the definition of the binomial coefficient to the case where k can be any real number. In this case, we're just going to have on the top k times k minus 1 multiplied by all the successive numbers
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down to k minus i plus 1. We're still going to divide by i factorial, so it's the same formula. It's just that we allow k to be any real number. Then the theorem states that 1 plus x to the k is equal
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to the sum from 0 to infinity of k choose n times x to the n. So this gives us the following infinite series. We can see using the ratio test that this series will always converge for x absolute value less than 1. So this
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formula holds in that case. For example, if we take k to be equal to minus 1, 1 plus x to the minus 1, it's going to be now we just take this formula and we put in minus 1 everywhere the k is, so we get
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minus 1 times x and then minus 1 times minus 2 over x squared, minus 1 minus 2 minus 3 times x cubed over 3 factorial, etc. And if you think about this, what we're getting up here is going to cancel
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out. It's essentially going to be either plus or minus n factorial divided by n factorial, so the coefficients, the denominator and the numerator cancel out and just leave us with an alternating sign. So we have 1 minus x plus x squared
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minus x cubed. Of course, this is exactly the formula that we obtained earlier from a completely different method by expressing, thinking of 1 plus x as 1 over 1 plus x is the sum of a geometric series. And if you remember, this equality only
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holds when x has absolute value less than 1, so there's no way that we could expect a better bound for the values of x for which this is true. And then x having absolute value less than 1. Let's
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talk briefly about where this formula comes from. Well, we know what the formula for a general Maclaurin series is. We need to find the nth derivative of the function and evaluate it at 0. So let's do this. If we differentiate this once, we
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get k times 1 plus x to the k minus 1. We do it again, we get k times k minus 1 times 1 plus x to the k minus 2, etc. So each time we reduce the power of 1 plus x by
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1. So if we do it n times, it goes down to k minus n, and each time we multiply by the current power of k, so we get a series of numbers k times k minus 1, etc., down to k minus n plus 1. When we
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evaluate at 0, the term 1 plus x just becomes one to some power, so that's one. So we're just left with this product of terms here. Now we insert this into the general formula for the Taylor series, so our coefficient of
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x to the n is this number divided by n factorial. And once we do divided by n factorial, we've got exactly this generalized binomial coefficient k choose n. So the result is that the Maclaurin series for this function is
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the sum from 0 to infinity k choose n times x to the n. We can see that this converges when x has absolute value less than 1 by the ratio test. It takes a little more work to show that actually
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converges to the original function. That's not guaranteed if you remember, but we want to address that question here. So let's move on and finish with one more example of how to use this formula. Let's take the case k was 1/2.
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This gives us the Maclaurin series for the square root of 1 plus x. So remember the general formula was this involving k. We just go ahead and insert everywhere there's a k, we replace it by 1/2, so we get this formula, and we
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think a little bit about what that is to simplify it. The 1/2 minus 1 becomes minus 1/2. The 1/2 minus 2 becomes minus 3/2, and then finally we simplify that again. So for instance, the x squared
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term is going to have 2 factorial on the bottom, then 2 times 2, which is 2 squared, and we have a negative sign. For the next coefficient, we have 2x minus 3 over 2, so that changes the minus to a plus, adds
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in a 3 on the top, and a 2 on the bottom, and the 2 factorial changes to 3 factorial. So if we continue with that pattern, we see that the next n will have 4 factorial on the bottom times 2 to the
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4th because we're multiplying by minus 5 over 2, so we'll also need to add a 5 on the top and change the sign. And if we continue with this, we see we're going to get the product of the
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successive odd numbers on the top and on the bottom in factorial times 2 to the n. So this is the kind of answer we get when we try to explicitly get formulas for 1 plus x to the k for
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different values of k. Generally, it'll be hard to give it in closed form as a summation, but it certainly should be easy like this to calculate the first four or five terms of the Maclaurin series.
Topics:binomial seriesMaclaurin seriesTaylor seriesbinomial coefficientsPascal's trianglebinomial theoreminfinite seriesseries convergenceNewton binomial theoremsquare root series
