General Form (MacLaurin Series)

zodac

Can someone explain how to get the general form for something like:
ln(1+x)=x - x^2/2 + x^3/3 - x^4/4....

I just can't get my head around it in my book and it hasn't come up for a while so I wanna b ready.

spoonbadger

zodac wrote: »

Can someone explain how to get the general form for something like:
ln(1+x)=x - x^2/2 + x^3/3 - x^4/4....

I just can't get my head around it in my book and it hasn't come up for a while so I wanna b ready.

Errr? .

U1 = x
U2 = x^2/2
U3 = x^3/3

The power and bottom number is the same as the term, so you get x^n/n

As for the power, it's (-1)^n+1. Might want to learn that little term for dealing with signs. Tends to be helpful .

BLARG

I have absolutely no grasp of "the general term"... you have some insane thing with loads of cosines and stuff, and then the question is "what's the general term" and it's just like n^2 +1 or something...

andyman

I'd look at sinx as well if I were you. That hasn't been up in 8 years.

EDIT: To answer your question. I know why they're the general term, but I can never, ever construct them myself. I'm just learning them from my book.

robz18

But Cosx came up just last year. I doubt Sinx will come up right after it.
My money is on either tan inverse or a log one.

Nimur90

Im just learning them off by heart now because I cant understand it either! theyre all in texts and tests 5

Piste

I would love if ex or sinx came up, but cosx came up last year and ex has come up a few times.


Hope to god they steer clear of the binomial. That came up two years ago so we should be same, but I will honestly die if tan inverse 1 comes up

Zonda999

Gonna learn them too. I know that i definitely wouldn't be able to form them in the exam. Horrible things to learn though.

I agree than an inverse tan one might come up. That said, we're probably due a max/min Q in the part C so who knows.

spoonbadger

Piste wrote: »

I would love if ex or sinx came up, but cosx came up last year and ex has come up a few times.


Hope to god they steer clear of the binomial. That came up two years ago so we should be same, but I will honestly die if tan inverse 1 comes up

I know it's not helpful, and just sounds narky, but seriously!!. Just learn them! .

I managed to go over the whole maclaurin series today in an hour (sorry, i know this sounds like an arrogant look-at-me-Im-an-A1-student rant already) despite having never understood it before. You can learn the tan-inverse, and binomial ones in about 20 mins. Give me a pm if you want some notes to look over! .

Piste

Thanks I did try to look at the binomial and tan, but they're so long it'll take me forever to learn them and I have to rest of paper 2 to study for (I'm trying to cut my losses). I'll just have to take a gamble and hope it pays off =/

It's simple enough once you get your head around it, really.
Using the above example:

log_e(1+x)= x - x^2/2! + x^3/3! - x^4/4! + ...

You've to do each little component individually.

First, look at the power, it's equal to the factorial of the divisor. So, you can take care of both of them at once. Now, nicely, you can see that the first term is to the power of one, second term is to the power of two, the third is to the power of three, etc.

So, x^n/n! will give each term.

If the first term was x^2 and not x, then, x^(n+1) would give you what you needed.

Now, look at the sign. The first term is positive, the second is negative, and this repeats itself.

(-1)^(n+1) will make the first term positive, the second negative etc.

Now, we just combine both: [(-1)^(n+1).x^n]/n!

It's easy if you can just break it up into different components.

Alexl

-JammyDodger- wrote: »

It's easy if you can just break it up into different components.

I agree, if you want a full proof way of getting the power right, its always an arithmetic series, X^ 0,2,4,6.......etc use arithmetic formula and then the factorial on the bottom is the same as the power of x

sock puppet

Piste wrote: »

Thanks I did try to look at the binomial and tan, but they're so long it'll take me forever to learn them and I have to rest of paper 2 to study for (I'm trying to cut my losses). I'll just have to take a gamble and hope it pays off =/

I have everything I need for binomial series written on a page. I'm gonna look at it before the exam and then do it first if it comes up. It should work. Hopefully anyway.

Zonda999

sock puppet wrote: »

I have everything I need for binomial series written on a page. I'm gonna look at it before the exam and then do it first if it comes up. It should work. Hopefully anyway.

Not a bad idea, that.I'd definitely forget the cos and sin general terms. I'm yet to learn them, though:rolleyes:

xOxSinéadxOx

I really don't get how you find the genral term, do they always ask it?

Nimur90

never have to look at the maclaurin series again!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1