The product of 2 infinite sums

In summary, the conversation discusses the product of two infinite series and how to find the result without using exponentials. Different methods, such as grouping products by degree of x and using symmetry properties, are suggested to achieve this.
  • #1
dyn
773
62
Hi.
I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
ΣnΣm xn/(n!) (-x)m/(m!)
without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ?
Thanks
 
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  • #2
What do you get, if you do not ignore the ##i## as you did, and group the products by degree of ##x##?
 
  • #3
dyn said:
Hi.
I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
ΣnΣm xn/(n!) (-x)m/(m!)
without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ?
Thanks
So the series for eu is Σn un/(n!)
and u is either ix or -ix (in your example). But you lost the i when converting to the series.
 
  • #4
Yes sorry I forgot the i when writing out the infinite series
 
  • #5
You may also use: ##e^{ix}=(cosx+isinx)## and symmetry properties of ##sinx, cosx##.
 
  • #6
WWGD said:
You may also use: ##e^{ix}=(cosx+isinx)## and symmetry properties of ##sinx, cosx##.
Sure, you could do that for this particular problem, but this ignores the OP's question about the product of two infinite series.
 

FAQ: The product of 2 infinite sums

1. What is the product of 2 infinite sums?

The product of 2 infinite sums is a mathematical concept that involves multiplying two infinite series together. An infinite sum is a series of numbers that continues infinitely, and the product of two infinite sums is the result of multiplying each term of one series with each term of the other series.

2. Can the product of 2 infinite sums be finite?

Yes, the product of 2 infinite sums can be finite. It depends on the convergence of the two infinite sums. If both series converge, the product will be finite. However, if one or both series diverge, the product will be infinite or undefined.

3. How is the product of 2 infinite sums calculated?

The product of 2 infinite sums can be calculated using the Cauchy product, which is a method for multiplying two infinite series. It involves multiplying each term of one series with each term of the other series and then summing the resulting products. This process continues until all possible combinations have been multiplied and added.

4. What is the significance of the product of 2 infinite sums in mathematics?

The product of 2 infinite sums has various applications in mathematics, including in calculus, number theory, and complex analysis. It is also used in solving differential equations and in the study of power series. Understanding the properties and behavior of the product of 2 infinite sums is crucial in many mathematical fields.

5. Can the product of 2 infinite sums be used to solve real-world problems?

Yes, the product of 2 infinite sums can be used to solve real-world problems, particularly in physics and engineering. It is used to approximate and model various phenomena, such as electrical circuits, fluid dynamics, and population growth. The product of 2 infinite sums allows for more precise and accurate solutions to these problems.

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