Converge or Diverge?

  • Thread starter JonF
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In summary, the convergence of the given sum \sum_{i=1}^{\infty} x^{C-i} depends on the absolute value of x. It can be rewritten as x^(C)*Sum(i=1,inf)1/(x^(i)), which is a geometric series in (1/x). This series converges for |1/x|<1, which means x< -1 or x> 1. The convergence or divergence of the sum is not affected by whether x is a whole number or not. The sum can also be expressed as x^C \sum_{i=1}^{\infty} x^{-i}, which simplifies as x^c/(x-1) or
  • #1
JonF
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Does this sum converge of diverge? C is a constant

[itex] \sum_{i=1}^{\infty} x^{C-i} [/itex]

Is there an easy way to tell if something converges or diverges?
 
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  • #2
Convergence of the given sum depends on the absolute value of x.

Note that you can rewrite your sum as x^(C)*Sum(i=1,inf)1/(x^(i))
But this is closely related to the well-known geometric series..
 
  • #3
Ok if X is a whole number, it diverges right?
 
  • #4
It is a geometric progression with initial term x^c and common ratio x^{-1} (arildno has a minus sign missing}

as such it converges for all reall x with |1/x|<1, ie |x|>1
 
  • #5
Since C is a constant and the sum is over the index i, you can take xC out of the sum and get
[itex] x^C \sum_{i=1}^{\infty} x^{-i} [/itex]

You should now be able to recognize the remaining sum as a geometric series in (1/x) which converges for -1< 1/x < 1- that is, x< -1 or x> 1.
Convergence and divergence has nothing whatsoever to do with whether x is a whole number or not.
 
  • #6
Ok here is my next question, is it possible that:

[itex]X^C < \sum_{i=1}^{\infty} X^{C-i} [/itex]
 
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  • #7
That would depend on what X, and x are. As the sum is [tex]\frac{x^{C+1}}{1-1/x}[/tex] given that for the sum to make sense |x|>1. I'm sure you can do the manipulation.

edited to allow for the sum running from 1 to infinity, not 0 to infinity
 
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  • #8
Hmm.., I thought the sum was x^(C)/(x-1)
 
  • #9
it could well be, i get bored keeping track of the details. i now think it is x^{c-1}/(1-1/x), which was the second one i put in there and works out at x^c/(x-1) doesn't it?
 
  • #10
Sure, I thought it was merely a typo or someting.
 

1. What does it mean for a series to "converge" or "diverge"?

When talking about a series, "convergence" refers to the idea that the terms in the series approach a finite value as the number of terms increases. In contrast, "divergence" means that the terms in the series increase without bound as the number of terms increases.

2. How can I determine if a series will converge or diverge?

There are a few different tests that can be used to determine the convergence or divergence of a series, such as the ratio test, the root test, or the integral test. These tests involve evaluating the limit of a certain function as the number of terms approaches infinity. If the limit is a finite number, the series converges. If the limit is infinity or does not exist, the series diverges.

3. What is the significance of convergence and divergence in mathematics?

The concept of convergence and divergence is essential in mathematics because it allows us to analyze infinite series and determine if they have a finite value. This is crucial in various applications, such as in physics, engineering, and economics, where infinite series are used to model real-world phenomena.

4. Can a series both converge and diverge?

No, a series can either converge or diverge, but not both. This is because the definition of convergence implies that the series approaches a finite value, while divergence implies that the series increases without bound. These two concepts are mutually exclusive.

5. Are there any real-life examples of series that converge or diverge?

Yes, there are many real-life examples of series that converge or diverge. One common example is the geometric series, which is used to calculate compound interest in finance. Another example is the harmonic series, which is used to analyze the convergence of alternating current circuits in electrical engineering.

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