The Binomial Theorem for Any Index

Transcript

0:00Introduction

Hello everyone! Okay, one strange puzzle first. You will soon see that one over, one minus x, equals one, plus x, plus x squared, and so on, forever. Now put x equal to two. The left side is one over minus one, that is, minus one. The right side is one, plus two, plus four, plus eight, and so on. Positive numbers adding up to minus one? Something is badly wrong, right? By the end of today's lesson on negative and fractional powers, you will know what went wrong. It is a handy tool for JEE problems, so let's begin.

0:41Any index: the series never stops

Now the index n can be negative, or a fraction. One plus x, to the power n, equals one, plus n x, plus n times n minus one, over two factorial, x squared, plus n times n minus one times n minus two, over three factorial, x cubed. The exclamation mark is factorial; the three dots mean the terms go on. Okay? T with r plus one written small below it is the general term, read T r plus one, counted from r equals zero. It is n times n minus one, down to n minus r plus one, over r factorial, times x to the power r. Right? Now think. Put n equal to three, and find the term in x to the power four. Zero, exactly! The top is three times two times one times zero, over four factorial; the raised dots mean times. So for a positive integer, a factor hits zero, and the series stops. Clear? But for n negative or a fraction, no factor is ever zero, so the series never stops. With me? And the price? It works for any rational n, but only when mod x is less than one. The two vertical bars mean modulus, the size of x without its sign. Got it?

1:57Series to know by heart, all for |x| less than 1

Now try the one on the badge with our formula: one minus x, to the power minus one; the small minus one up there is the index. So n is minus one, and x is replaced by minus x. Find the first three terms. Let's check. n x is minus one times minus x: plus x. Then minus one times minus two, over two, is one: plus x squared. Every coefficient is one! With me? Change minus x to plus x, and the signs alternate: one minus x plus x squared, and so on. Okay? Next, one minus x, to the power minus two: one plus two x plus three x squared, and so on. The big Greek letter sigma means sum: the sum of r plus one, times x to the power r. Right? And to the power minus three: the sum of r plus one, times r plus two, over two, times x to the power r. Clear? Now the master formula. One minus x, to the power minus n, is the sum, for r greater than or equal to zero, of n plus r minus one C r, x to the power r. The curly E sign means belongs to, and the double struck capital N is the set of natural numbers, so n is natural here. Got it? Now a quick check. Put n equal to two in the master formula. Does it match the third line? It does! n plus r minus one C r becomes r plus one C r, which is just r plus one: the third line. And remember, as the heading says, all these series need mod x less than one. Okay?

3:34The trap: valid only when |x| is less than 1

Now the biggest warning. For a positive integer n, the series is finite and valid for every x. But our puzzle: x equal to two in one over, one minus x, gave minus one equals one plus two plus four, and so on. Where did it go wrong? Yes, the condition! For any other index, the series is infinite, and valid only for mod x less than one. Outside it, the series diverges and the formula is false. Clear? And x equal to two is far outside. So one plus two plus four, and so on, is not minus one; the slashed equals sign means not equal to. Mystery solved! Always check the condition first. Okay? What if the bracket does not start with one? For a plus b, to the power n, take out a: a to the power n, times one plus b over a, to the power n. So now we need mod of b over a less than one. With me?

4:35Small x: quick approximations

Now a real time saver. When x is small, x squared and higher powers are tiny. So one plus x, to the power n, is approximately one plus n x; the wavy equals sign means approximately. That is first order. Right? Need more accuracy? Keep the x squared term too: plus n times n minus one, over two, x squared. That is second order. Clear?

5:01Example 1: expand (1+x)^(1/2) up to x^3

Now, Example one. Expand one plus x, to the power one half, up to the term in x cubed; the one slash two up there is one half. Pause the video and try it. Done? Okay, let's check. Here n is one half, so n minus one is minus one half, and n minus two is minus three over two. Write these first; it saves sign mistakes. Okay? The x term is n times x, which is one half x. Easy, right? Now the x squared term: n times n minus one, over two factorial. Before you calculate, positive or negative? Negative! One half times minus one half is minus one quarter; over two factorial, which is two, that is minus one eighth. So minus x squared over eight. Clear? Now x cubed: one half, times minus one half, times minus three over two, over three factorial, which is six. Two minus signs make it positive: three over eight, over six, is one over sixteen. So plus x cubed over sixteen. With me? So the answer: one, plus x over two, minus x squared over eight, plus x cubed over sixteen, and so on, valid for mod x less than one. Got it?

6:19Example 2: coefficient of x^5 in (1-2x)^-3

Now Example two, a little harder. Find the coefficient of x to the power five in one minus two x, to the power minus three. Pause the video and try it. Done? Okay, let's check. Here we have two x, so let y equal two x. With n equal to three, the master formula gives r plus two C r, the same as r plus two C two. So one minus y, to the power minus three, is the sum of r plus two C two, y to the power r. Okay? We want the fifth power, so r is five. Seven C two, which is twenty one. Clear? So the answer is twenty one, right? Oh no, careful! That is the coefficient of y to the power five, not x to the power five. And y to the power five is two x to the power five, that is, thirty two x to the power five. Got it? So the coefficient of x to the power five is twenty one times thirty two: six hundred and seventy two. Forgetting the thirty two is a common trap. With me?

7:24Quick revision: when it is valid, (1-x)^-n, taking out a^n

So, quick revision. One: if the index is not a positive integer, the series is infinite, valid only for mod x less than one. Two: one minus x, to the power minus n, is the sum of n plus r minus one C r, x to the power r, for natural n. Three: for a plus b, to the power n, take out a to the power n, and check mod of b over a is less than one. Now try the practice problems on this page. All the best!