Question

use (1+x)^p= 1+px+p(p-1)/2!x^2+ p(p-1)(p-2)/3!x^3+... to find the first four nonzero terms of the taylor series about 0 for arcsinx

Answer 1

hey u found me lol

Answer 2

Haha yeah.

Answer 3

Can you find the derivative of arcsinx for me?

Answer 4

I'll help you with the idea of this question, and then you try and let me know if you face any problem.

Answer 5

\[{d \over dx}\sin^{-1}x={1 \over \sqrt{1-x^2}}=(1+(-x^2))^{1/2}\]
Which is in the given form, after substituting with p=1/2

Answer 6

Now find the first 4 terms of the expansion of the derivative of arcsinx using the given formula you wrote above. After that, just integrate these four terms, and that would be your first five term of the taylor series of arcsinx.

Answer 7

You following?!

Answer 8

Hello?! :)

Answer 9

so do i plug in (-x)^2 in for x in the formula

Answer 10

Exactly, and 1/2 for p.

Answer 11

simple. right?!

Answer 12

then i should find the antiderivative to get the orginal

Answer 13

Yep!

Answer 14

Does that make sense?

Answer 15

yah just a general question when do first use the derivative to find the taylor series

Answer 16

Well, that depends on the question. In this case, the question tells us what method to use to find the Taylor expansion of arcsinx. Here we should look for anything related to arcsin, that's similar to the given formula. Its derivative is the best choice, I believe.

Answer 17

okay fantastic ty you for your help.

Answer 18

You're welcome!