Sum of Geometric Series: Ʃ(3→∞) 3(.4)^(n+2)

In summary, the sum of Ʃ(3→∞) 3(.4)^(n+2) can be found by using the formula for the sum of a geometric series with the first term being .03072 and the common ratio being 2/5. This results in a final sum of 32/625 or .0512.
  • #1
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Homework Statement



Find the sum of Ʃ(3→∞) 3(.4)^(n+2)

Homework Equations



Sum of Geometric Series = ao/(1 - r), ao=3, r = .4 = 2/5

The Attempt at a Solution



I thought that I could use the definition of the sum of a geometric series (above) to determine the sum of this equation. So 3/(1-.4) = 5. I believe that somewhere I need to account for the difference in taking the sum of the series from 1 (like I normally would) and 5 (series starts at 3 and the ratio is n+2). But I'm not sure how I do that.
 
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  • #2
First off, what's the first term of the series? (a is not 3)

Next, r is what you multiply by to get from one term to another. Write out the first few terms of the series. What do you multiply by to get from the first term to the second?

Now that you have a and r, you can calculate the sum.
 
  • #3
Ah! The first term of this series would be .03072.

I can only use a0 = 3 if I started this series at n=1. I have to recalculate my a0 term given the changed starting point.

This series converges to 32/625 or .0512.

Thanks!
 

1. What is the formula for the sum of a geometric series?

The formula for the sum of a geometric series is Ʃ(3→∞) ar^(n-1), where a is the first term and r is the common ratio.

2. How do you find the sum of a geometric series?

To find the sum of a geometric series, use the formula Ʃ(3→∞) ar^(n-1), where a is the first term and r is the common ratio. Plug in the values of a and r and solve for the sum.

3. What does the notation "Ʃ(3→∞)" mean?

The notation "Ʃ(3→∞)" represents the summation notation, where the starting index is 3 and the ending index is infinity. This means that the sum will include all terms from the starting point (3) to infinity.

4. Can you calculate the sum of a geometric series with a finite number of terms?

Yes, the sum of a geometric series can be calculated with a finite number of terms by using the formula Ʃ(3→n) ar^(n-1), where n is the number of terms to be included in the sum.

5. How is a geometric series used in real life?

A geometric series can be used in real life to calculate compound interest, population growth, and depreciation of assets. It is also used in physics and engineering to model exponential decay and growth.

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