How to determine this infinite sum

In summary, the value of an infinite sum can be determined by using a limit, which involves adding up an increasing number of terms and finding the value that the sum approaches as the number of terms approaches infinity. The formula for determining an infinite sum is S = a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio between each term. An infinite sum can have a finite value if it converges, and there are tests such as the comparison test, ratio test, and integral test to determine if it converges or diverges. An infinite sum can also have a negative value if it is a convergent alternating series, with terms that alternate between positive and negative values and approach
  • #1
Hi,

How do i determine de result in terms of x of this series:

(Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)
For x<1

Thanks
 
Last edited:
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  • #2
The denominator is a simple geometric series = 1/(1-x). The numerator series can be gotten using elementary calculus from the denominator series = xd/dx(1/(1-x))=x/(1-x)^2.
Net result = x/(1-x).
 

1. How do you determine the value of an infinite sum?

The value of an infinite sum can be determined by using a mathematical concept called a limit. This involves adding up an increasing number of terms in the sum and finding the value that the sum approaches as the number of terms approaches infinity.

2. What is the formula for determining an infinite sum?

The formula for determining an infinite sum is S = a + ar + ar^2 + ar^3 + ..., where a is the first term in the sum and r is the common ratio between each term.

3. Can an infinite sum have a finite value?

Yes, an infinite sum can have a finite value if the sum converges. This means that the value of the sum approaches a finite number as the number of terms approaches infinity.

4. How do you know if an infinite sum converges or diverges?

There are various tests that can be used to determine if an infinite sum converges or diverges. These include the comparison test, the ratio test, and the integral test. These tests involve evaluating certain properties of the terms in the sum to determine if the sum will approach a finite number or if it will continue to increase indefinitely.

5. Can an infinite sum have a negative value?

Yes, an infinite sum can have a negative value if the sum is a convergent alternating series. This means that the terms in the sum alternate between positive and negative values, eventually approaching a finite number. An example of this is the sum S = 1 - 1/2 + 1/3 - 1/4 + ..., which has a finite value of ln(2).

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