Question

Can you please help me I do not know how to simplify the infinite product of (1+x)(1+x2)(1+x4)(1+x8)(1+x16).., given |x| < 1. Please help....

Answer 1

consider the partial products :
n=1 : 1+x
n=2 : (1+x)(1+x^2) = 1+x+x^2 + x^3
...the pattern continues
n=k : 1+x+x^2 + ... x^(2^k-1)

Answer 2

express it as partial sum for n=k, and take the limit \(k \to \infty \)

Answer 3

hmmmm i'm not following you

Answer 4

maybe some binomial theorm will help ?

Answer 5

not binomial itself , the inverse of it

Answer 6

for n = k :

\[\large 1+x+x^2 + \cdots x^{2^k-1} = \dfrac{1-x^{2^k}}{1-x}\]

Answer 7

ok...

Answer 8

does above sum looks okay ? :)

Answer 9

|x|<1 is givven for a reason :D

Answer 10

yeah

Answer 11

it is correct

Answer 12

the key thing to notice in the start is this :
(a+b)(c+d) gives you 2*2 = 4 = 2^2 terms
(a+b)(c+d)(e+f) gives you 2*2*2 = 8 = 2^3 terms

Answer 13

as you increase the number of binomials in ur product, the terms in expansion increases in powers of 2

Answer 14

oooookkkkkk.....

Answer 15

take the limit for the infinite product :

\[\large 1+x+x^2 + \cdots x^{2^k-1} + \cdots = \lim \limits_{k\to \infty } \dfrac{1-x^{2^k}}{1-x}\]

Answer 16

ohhhh i see. Its getting a bit clearer now

Answer 17

since |x| < 1, \(x^{2^k} \to 0 \) as \(k \to \infty \)
@Somy

Answer 18

i roger that xD
i only need to get back to my acount to loo smart in math :P

Answer 19

ok i think i can n from here but is it ok if you wait and check if i got the right answer?

Answer 20

sure :)

Answer 21

how did u manage to get into all these accounts ? are you the one using Kainui's account also ? :o

Answer 22

ok

Answer 23

i believe the answer is\[\frac{ 1 }{ 1-x }\]

Answer 24

am i correct?

Answer 25

Looks good ^^

Answer 26

you sure because i don't want to get it wrong

Answer 27

oh well ill take that as a yes

Answer 28

It's right

Answer 29

ok thanks @ganeshie8 @Astrophysics @Somy

Answer 30

you were great help!!!

Answer 31

np :)

Answer 32

Now this will be a Chating room!!!

Answer 33

chatting*

Answer 34

jk

Answer 35

see ya suckas