Application of the Cauchy product

[naaa00]

Homework Statement

Hello,

I'm trying to find the Taylor representation of a product of functions - the exponential of x times sin(y). Also for (x - y)sin(x+y).

The Attempt at a Solution

Well, I want to use the Cauchy product in both cases.

I know the taylor representation of both functions:

Ʃ x^n / n! = e^x (a_n)

Ʃ (-1)^k * y^(2k+1) / (2k+1)! = sin(y) (b_k)

(infinite sums)

so:

Ʃ x^n / n! * Ʃ (-1)^k * y^(2k+1) / (2k+1)!

the Cauchy product:

ƩƩ [ x^n / n! * (-1)^(n-k) * y^(2(n-k)+1) / (2(n-k)+1) ] (Both sums got to infinity)

Here I'm stuck. I don't know what I could do to simplify that expression. Any suggetions?

For the other example, is it valid to use the taylor representation of the sine function and just substitute the argument with (x+y) ? And then multiply the (x-y) with the sum?

Thanks!

 

[mighty2000]

Hello,

Yes, it is valid to use the Taylor representation of the sine function and substitute the argument with (x+y). This will give you the Taylor representation for (x+y)sin(x+y). To find the Taylor representation for (x-y)sin(x+y), you can use the same approach and substitute the argument with (x+y) - 2y. This will give you the final expression for the Cauchy product of (x-y)sin(x+y).

As for the first example, you are on the right track by using the Cauchy product. However, you can simplify the expression further by using the properties of exponents and factorials. For example, you can write x^n * (-1)^(n-k) as (-1)^k * x^(n-k). This will help you simplify the expression and find the Taylor representation for the product of exponential of x and sine of y.

I hope this helps. Let me know if you have any further questions.