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Chapter 11: Problem 94

How do you determine if an infinite geometric series has a sum? Explain how to find the sum of such an infinite geometric series.

Short Answer

Expert verified
An infinite geometric series has a sum or converges if the absolute value of the common ratio 'r' is less than 1. The sum 'S' of the series can be calculated using the formula S = a / (1 - r), where 'a' is the first term of the series. If the absolute value of the common ratio 'r' is greater than or equal to 1, the series diverges and does not have a sum.

Step by step solution

01

Identify a and r

First, identify the first term 'a' and the common ratio 'r' of the series. The first term 'a' is the first value in the series. The common ratio 'r' is found by dividing the second term by the first term.
02

Check for converge or diverge

Next, determine whether the series converges or diverges. If the absolute value of 'r' is less than 1, then the series converges. But, if the absolute value of 'r' is greater than or equal to 1, then the series diverges.
03

Calculate the sum of the series

If the series converges, calculate the sum of the series using the formula S = a / (1 - r). It's important to note that the sum is only valid for a converging series.

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Most popular questions from this chapter

Suppose that it is a drawing in which the Powerball jackpot is promised to exceed \(\$ 700\) million. If a person purchases \(292,201,338\) tickets at \(\$ 2\) per ticket (all possible combinations), isn't this a guarantee of winning the jackpot? Because the probability in this situation is \(1,\) what's wrong with doing this?

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.
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