Question

Some one explain please
For the validity of a binomial expansion[ (1+x)^n], some books say its valid if |x|>1 and some books say valid if -1>x>1 what do they both mean and are they the same?

Answer 1

well, the binomial theorem works with exponents that are integer
now if you have a negative exponent that's an integer, well \(\bf a^{-n} = \cfrac{1}{a^n}\qquad \qquad (x+y)^{-n} = \cfrac{1}{(x+y)^{n}}\)

Answer 2

Yeah but what does -1>x>1 mean in math notation?

Answer 3

And does |x|>1 mean only fractions?

Answer 4

|x|>1 and -1>x>1 are the same thing :)

Answer 5

well, -1 > x # "x" is less than -1
keep in mind that on the negative values, the farther from the 0, the smaller, thus
-25 is smaller than -1

-1 > x # will mean \(\bf (-\infty, 1)\)

Answer 6

@Zepdrix But -0.5 can fall in range range of -1>x>1 but not |x|>1

Answer 7

but -0.5 is not an integer

Answer 8

though ... -0.5 is greater than -1, and outside the range indicated

Answer 9

Sorry i've been using the wrong sign I mean |x|<1 and -1<x<1

Answer 10

Oh I'm sorry, I misread what your wrote.

I meant to say, -1>x>1 is equivalent to |x|<1

Not greater than.

Answer 11

Oh lol

Answer 12

hmmm, actually I have it right, even with the > lhehehhe

Answer 13

well, -1 < x # "x" is greater than -1
keep in mind that on the negative values, the farther from the 0, the smaller, thus
-25 is smaller than -1

-1 < x # will mean \(\bf (-\infty, 1)\)

Answer 14

hmmm, wait.... a sec

Answer 15

any greater .... will be to the rigtht... ok ... well have, better read closely

Answer 16

hmm |x|<1 and -1<x<1 that only leaves room for rationals

Answer 17

maybe they mean \(\bf a^{-n} = \cfrac{1}{a^n}\qquad (x+y)^{-n} = \cfrac{1}{(x+y)^{n}} \qquad \large (x+y)^{\frac{a}{b}} = \sqrt[b]{(x+y)^a}\)

Answer 18

Yes yes but why does -0.5 fall in range of -1<x<1 but not |x|<1

Answer 19

They mean when n is not a positive integer

Answer 20

@jdoe0001

Answer 21

-0.5 DOES fall in the range of |x|<1

It didn't when you had the equality sign flipped, but it does now :O

Answer 22

\[\Large |-0.5|<1 \qquad\to\qquad 0.5<1\]

Answer 23

Ohh yeeeeeaaaahh!! Sorry

Answer 24

XD

Answer 25

Thanks you both, can't give two medals so I won't give any X_X

Answer 26

I have too many medals XD you can give it to jdoe if you want. hehe

Answer 27

Can I have some help here

Answer 28

Letf(x)= 9x2+4 . (2x + 1)(x − 2)2
(i) Express f(x) in partial fractions.


(ii) Show that, when x is sufficiently small for x3 and higher powers to be neglected,
f(x) = 1 − x + 5x2.

What does it mean to show that x is sufficiently small for x^3 ?