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AP Calculus 1.1 Sequences and Series 237 Views


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AP Calculus 1.1 Sequences and Series. Find the Maclaurin series for the equation.


Transcript

00:00

Thank you We sneak and here's your shmoop du jour

00:05

Brought to you by scottish mathematician colin mclaurin Few men

00:09

have ever looked better in a dress Find the maclaurin

00:13

series for e f of x equals x squared over

00:18

one plus x And here the potential answers No What

00:23

a mess All right first of all what the heck

00:26

is the maclaurin series No not exactly Well in calculus

00:31

the maclaurin series is the expansion of a function that

00:34

revolves around x equals zero So this is the formula

00:39

f zero plus f prime of zero x plus f

00:44

double prime of zero over to factorial times x squared

00:48

et cetera You get our drift here see the pattern

00:51

We could make a tail and calculate the derivatives evaluated

00:54

at zero to find the maclaurin series of function given

00:57

But that would take some time And we're busy people

01:00

We've got laundry to do a much more efficient ways

01:02

to notice that x squared over one plus x equals

01:06

x squared times one over one minus negative x Well

01:10

we also know where we should know that The maclaurin

01:12

series for one over one minus acts is the summation

01:16

Of x to the power from any equal zero to

01:19

infinity so substituting the negative acts into the summation notation

01:23

and multiplying it by x squared we get x squared

01:27

times The summation of negative x to the power from

01:30

n equals zero to infinity still doesn't look like any

01:34

of our answer choices so let's simplify it even more

01:37

We can pull a negative one to the end out

01:40

of the negative expedient and we can combine the x

01:43

squared to the x to the end with the first

01:46

basic exponents room Whenever you multiply to terms with the

01:50

same base you can add the exponents Remember that Well

01:54

we're left with negative one to the end and x

01:57

to the power of end plus too so we're going

02:00

with answer joyce b o sounds like our laundry's done 00:02:03.879 --> [endTime] well we'll get it tomorrow Oh

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