How to calculate infinite series?

In summary, to calculate the series Ʃ(2^n * x^n), n starts from 0 to infinity, you can transform it into the classical geometric series Ʃ(X^n) where X = 2x. However, keep in mind that this transformation only works if |X| < 1.
  • #1
Hi, I'm stuck on a problem because I don't know how to calculate this series:
Ʃ(2^n * x^n), n starts from 0 to infinity. How can I calculate this? Do I have to transform it into something else where I know the outcome of the Σ like we do with the geometric functions?
 
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  • #2
thecaptain90 said:
Hi, I'm stuck on a problem because I don't know how to calculate this series:
Ʃ(2^n * x^n), n starts from 0 to infinity. How can I calculate this? Do I have to transform it into something else where I know the outcome of the Σ like we do with the geometric functions?

Let X=2x
Ʃ(X^n) is the classical geometric series.
 
  • #3
Thanks, didn't think of doing this.
 
  • #4
Be advised that |X| < 1 to treat this as a geometric series.
 

1. What is an infinite series?

An infinite series is a sum of an infinite number of terms. It can be represented as a1 + a2 + a3 + ..., where an is the nth term of the series.

2. How do you calculate the sum of an infinite series?

The sum of an infinite series can be calculated using various methods, such as the geometric series formula, telescoping series, or the ratio test. It is important to determine if the series converges or diverges before attempting to calculate the sum.

3. What is the difference between a convergent and divergent series?

A convergent series is one in which the sum of all the terms approaches a finite number as the number of terms approaches infinity. In contrast, a divergent series is one in which the sum of all the terms either approaches infinity or does not have a finite limit as the number of terms increases.

4. How do you determine if an infinite series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of an infinite series, such as the comparison test, integral test, and p-series test. These tests evaluate the behavior of the terms in the series and determine if the series will ultimately approach a finite sum or not.

5. Can an infinite series have a sum if it diverges?

No, an infinite series cannot have a finite sum if it diverges. This means that the sum of all the terms in the series either approaches infinity or does not have a finite limit. In this case, the series is said to have no sum or to be divergent.

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