Question

Cauchy Product

Answer 1

show that the cauchy product of the series

\[1 + \sum_{n=1}^{\infty}{\frac{(-1)^{n-1}}{2n} \frac{ ((1-\frac{1}{2})(2-\frac{1}{2})...((n-1)-\frac{1}{2})}{(n-1!)}}x^n\]

Answer 2

and

\[1 + \sum_{n=1}^{\infty}{\frac{(-1)^{n}(1-\frac{1}{2})...(n-\frac{1}{2})}{n!} x^n}\]

is equal to 1

Answer 3

It's probably just a matter of using the definition of Cauchy product.

Answer 4

that's what i thought but it wont come out.. i have been trying for a while

Answer 5

What is your definition of Cauchy product?

Answer 6

http://en.wikipedia.org/wiki/Cauchy_product

Answer 7

Is the Cauchy produce the actual term \(c_n\) or the result of the infinite sum or ...?

Answer 8

it is the sum

Answer 9

it seems that the first term of the product is 1 and all others go to zero, but i'm sturggling to show this in general

Answer 10

What do you get when you multiply them out?

Answer 11

i tried to write it out but the equation writer keeps crashing. i'll have another go at this problem in the morning. it should work out because the first series is for sqrt(1+x)

and the second is for 1/sqrt(1+x) . i cant use this fact though.