Question

Expand the following expression as a series in ascending powers of x up to the terms in x^3. State the range of values of x in each case for which expansion is valid.

Answer 1

@ganeshie8

Answer 2

@phi

Answer 3

is this calculus? and do they expect a taylor series expansion?

Answer 4

Binomial expansion

Answer 5

are you sure? The binomial expansion tells us how to expand forms
(a+b)^n

Answer 6

Sorry, it's binomial series

Answer 7

oh. According to https://en.wikipedia.org/wiki/Binomial_series
that means taylor series about x=0 of a function in the form
\[ f(x) = (1+x)^\alpha \]

so the first step is write your expression in that form

Answer 8

The person who taught me.. had left..
He said I can use this
http://prntscr.com/49rlmz

Answer 9

yes, but we need to write you expression in that form. Let me think about it.

Answer 10

ok

Answer 11

Here is the idea. write your problem as
\[ (x+2) ( 1 - 3x)^{-\frac{1}{2}} \]

now expand only
(1-3x)^ (-½) up to x^3
using
https://en.wikipedia.org/wiki/Binomial_series

then multiply that by (x+2) , collect terms, and keep only up to x^3 terms

Answer 12

can you expand (1-3x)^ -½ ?

Answer 13

I'm stuck at this part

Answer 14

Here is the formula

what is "alpha" for your problem?
what is "x" for your problem?

Answer 15

in other words match up
\[ (1+x)^\alpha \]
and
\[ (1+ - 3x ) ^ {- \frac{1}{2}} \]

do you see alpha is -½ ?
and that x in the formula matches (-3x) ?