Series Expansions

Transcript

0:00Introduction

Hello everyone! Okay, take your calculator, set to radians. Sine of zero point one is zero point zero nine nine eight three, and more digits. Very close to zero point one, right? So near zero, sine x is almost x. But almost is not equal. The gap, x minus sine x, is about zero point zero zero zero one six seven. Tiny, but not zero. Where does that gap come from? Today's tool, series expansions, tells us exactly. And it cracks limits that come up often in JEE, where the standard results are not enough. So, let's begin.

0:38Expansions near x=0

So, the big idea. Near x equals zero, each function becomes a polynomial that never ends. First, e to the power x; e is the special number about two point seven one eight. E to the power x equals one, plus x, plus x squared over two factorial, plus x cubed over three factorial; the three dots mean the terms go on forever, okay? The exclamation mark means factorial: two factorial is two, three factorial is six. With me? Next, l n of one plus x; l n is the natural log, log to base e. Log of one plus x equals x, minus x squared over two, plus x cubed over three, and so on. Signs alternate, and look: no factorials here. Careful, students mix these up. Right? Now, sine x equals x, minus x cubed over three factorial, plus x to the power five over five factorial, and so on. Only odd powers, signs alternating. Clear? Now cos x. Sine has only odd powers. So what does cos x have? Even powers, exactly! Cos x equals one, minus x squared over two factorial, plus x to the power four over four factorial, and so on. Got it? And tan x equals x, plus x cubed over three, plus two x to the power five over fifteen, and so on. Look: a plus sign before x cubed over three, not minus like sine. Remember that. Okay?

2:11More expansions near x=0

Now, some more from the course. One plus x, whole to the power n, equals one, plus n x, plus n times n minus one, over two, times x squared, and so on. Right? Then the inverses; the small minus one up top means inverse, not one over. Sine inverse x equals x, plus x cubed over six, and so on. Tan inverse x equals x, minus x cubed over three, and so on. With me? Next, for a positive number a, a to the power x equals one, plus x log a, plus x log a whole squared, over two, and so on. It is just the e to the power x series, with x log a in place of x. Clear? And the last one: one plus x, to the power one over x, equals e times, in brackets, one, minus x over two, plus eleven x squared over twenty four, and so on. Same e as before, the base of the natural log. Okay?

3:11The rule: match the power in the denominator

Now, how do we use these in a limit? One rule. Expand every term up to the lowest power of x in the denominator. Bottom is x cubed? Go up to x cubed. With me? Course example: the limit as x tends to zero, of log of one plus x, minus x, over x squared; lim is short for limit, and the arrow means tends to. It is a zero over zero form. Up to which power do we expand log of one plus x? Up to x squared, correct! Subtract x from the series: the x terms cancel, leaving minus x squared over two, plus x cubed over three, and so on. Clear? Divide by x squared: minus one half, plus x over three, and so on. Every later term tends to zero, so the limit is minus one half. Higher powers simply vanish. Got it?

4:04Topper's trick: the four cubic terms

Now, a topper's trick: learn the four cubic terms as two pairs. The wavy equals sign means approximately. Sine x is about x minus x cubed over six, since three factorial is six. Sine inverse x: the same six, opposite sign. Okay? And tan x is about x plus x cubed over three. Tan inverse x: the same three, opposite sign. Right? Course example: sine inverse x, minus tan inverse x, over x cubed, as x tends to zero. What is the limit? One half, very good! The x terms cancel, and x cubed over six, plus x cubed over three, is x cubed over two. Divide by x cubed: one half. That's it!

4:55Example 1: limits of x - sin x and tan x - x over x cubed

Now, Example one, an easy one from the course. Find the limit as x tends to zero, of x minus sine x, over x cubed. And also the limit of tan x minus x, over x cubed. Pause the video, and try it first. Done? Okay, let's check. Near zero, sine x is almost x. So the top is x minus x, zero, and the limit is zero, right? Oh no, careful! We stopped at x, but the bottom is x cubed. Our rule says: expand sine x and tan x up to x cubed. Clear? So, x minus sine x equals x, minus, in brackets, x minus x cubed over six, and so on. The x terms cancel, leaving x cubed over six. Divide by x cubed: one sixth. And that is our puzzle solved: the tiny gap, x minus sine x, is about x cubed over six! Now, tan x minus x is x cubed over three, and so on. Divide by x cubed: one third. So, one sixth and one third. Very good!

6:08Example 2: find a and b using expansions of sin x and cos x

Now a harder one from the course, of a type that comes up often in JEE, okay? Find a and b, if the limit as x tends to zero, of x times one plus a cos x, minus b sine x, all over x cubed, equals one. Pause the video and try. Hint: expand up to x cubed. Done? Okay, let's check. We need the top up to x cubed. Cos x is already multiplied by x, so it needs only one minus x squared over two. So the top is x, plus a x times one minus x squared over two, minus b times x minus x cubed over six, and so on. With me? Now collect the powers. The x terms give one plus a minus b, times x. The x cubed terms give b over six, minus a over two, times x cubed. Right? Now think. The x term over x cubed is one plus a minus b, over x squared, and one over x squared blows up as x tends to zero. But the limit is just one. So what must one plus a minus b be? Zero, exactly! One plus a minus b equals zero, so b equals a plus one. Clear? Then only the x cubed term is left, and its coefficient must equal the limit, one. So b over six, minus a over two, equals one. Multiply by six: b minus three a equals six. With me? Now put b equals a plus one: a plus one, minus three a, gives one minus two a, equal to six. So a is minus five over two, and b is minus three over two. Two unknowns, two conditions, one from each power. Perfect!

7:53Quick revision: key expansions and the matching-power rule

So, quick revision. One: learn the expansions of e to the power x, log of one plus x, sine x, cos x and tan x, near x equals zero. Two: expand every term up to the lowest power of x in the denominator. Three: with unknown constants, set the lower coefficients equal to zero, then match the limit. Now go and try the practice problems on this page. You will enjoy them. All the best!