Infinite geometric series

In summary, an infinite geometric series is a series of numbers where each term is multiplied by a constant ratio to get the next term, and it can either approach a finite limit or diverge to infinity. The formula for finding the sum of an infinite geometric series is S = a / (1 - r), where a is the first term and r is the common ratio between terms. To determine if the series converges or diverges, the absolute value of the common ratio is compared to 1. And yes, an infinite geometric series can have a negative common ratio. In real-life applications, these series are used to model exponential growth and decay, as well as in finance and economics for calculating compound interest.
  • #1
Niaboc67
249
3
x+x^2+x^3+x^4... = 14

Find x

Could someone please provide an explanation on how to solve this?
 
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  • #2
The formula for infinite geometric series is ##\displaystyle \sum_{n=0}^\infty a_n x^n =\frac{a_0}{1-x} ##. But this is true only for ## -1 < x < 1##. Just use this on the series to get an equation in a familiar form.
 
  • #3
Same question asked in another thread with the same name, so locking this thread.

@Niaboc67, please don't start multiple threads on the same topic.
 

1. What is an infinite geometric series?

An infinite geometric series is a series of numbers where each term is multiplied by a constant ratio to get the next term. This series continues infinitely, and can either approach a finite limit or diverge to infinity.

2. What is the formula for finding the sum of an infinite geometric series?

The formula for finding the sum of an infinite geometric series is S = a / (1 - r), where a is the first term and r is the common ratio between terms.

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

If the absolute value of the common ratio (r) is less than 1, then the series will converge to a finite limit. If the absolute value of r is greater than or equal to 1, then the series will diverge to infinity.

4. Can an infinite geometric series have a negative common ratio?

Yes, an infinite geometric series can have a negative common ratio. The series will still converge or diverge based on the absolute value of the ratio.

5. How is an infinite geometric series used in real-life applications?

Infinite geometric series can be used to model exponential growth or decay, such as in population growth, interest rates, and radioactive decay. They can also be used in finance and economics, such as in calculating compound interest.

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