# Cauchy Product: Exponents or Counters Equal?

• morenogabr
In summary, the Cauchy product theorem is more important when the terms of the exponent of the x variable are the same.f

#### morenogabr

When attemtpting to find the Cauchy product of two functs f(x) and g(x), which are themselves power series, is it more important to have their respective terms of summation be the same of for the exponent of their respective x variable to be equal? Or must both of these conditions be met? I am in a situation where it seems that shifting the index for f(x) to make x powers equal, makes the initial summation counters not equal. Not sure which of these conditions is needed to apply Cauchy prod theorem.

why not show us what you are attempting, try click on the latex below to see the syntax
$$f(x) = \sum_{n=1}^{\infty} a_n x^n$$

as a general rule I woudl try & make teh summation powers of x equivalent. If the other subscripts are confusing why not try a variable change $c_n = b_{n-2}$ etc.

$$f(x) = e^x = \\sum_{n=0}^{\\infty} (1/n!)(x^n),$$
$$g(x) = sin(x) = \\sum_{k=0}^{\\infty} ((-1)^k)/(2k+1)! * x^(2k+1)$$

latex not working? maybe its clear enoug what I am trying to show...

I need to use cauchy product theorem to obtain the first 3 terms of f(x)g(x).
So for g(x) I say m=2k+1, k=(m-1)/2, etc to achieve x^m but then the sum goes from m=1 to inf.
So if I leave x powers equal, the summations start at 0 for f(x) and 1 for g(x). But the theorem only gives
c_n=sum(from 0 to inf) a_(k) * b_(n-k) not sure if this would need to go from 1 to inf in my case or leave as 0?

$$f(x) = e^x = \\sum_{n=0}^{\\infty} (1/n!)(x^n),$$
$$g(x) = sin(x) = \\sum_{k=0}^{\\infty} ((-1)^k)/(2k+1)! * x^(2k+1)$$

latex not working? maybe its clear enoug what I am trying to show...

I need to use cauchy product theorem to obtain the first 3 terms of f(x)g(x).
So for g(x) I say m=2k+1, k=(m-1)/2, etc to achieve x^m but then the sum goes from m=1 to inf.
So if I leave x powers equal, the summations start at 0 for f(x) and 1 for g(x). But the theorem only gives
c_n=sum(from 0 to inf) a_(k) * b_(n-k) not sure if this would need to go from 1 to inf in my case or leave as 0?

$$f(x) = e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$
$$g(x) = sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!}$$

Use this format for your latex, it's better.