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How can an infinite series have a finite sum? - Answers
It is possible for an infinite series to have a finite sum if it is of a "special" kind. Consider the infinite series of fractions that is 1/2, 1/4, 1/8, 1/16,...1/n. You can see that each term is 1 over twice the denominator of the previous term. (We might have written the last term as 1/2n where n is the number of the term.) In this series there are an infinite number of elements, but the sum will be 1. In the case of the series, 1/2 + 1/4 + 1/8 + 1/16 +... + 1/n, the sum has a limit of 1, and 1 will be the sum of the terms at infinity. Let's do the math. Add the first two terms, get a sum, than add the next term to get another sum, then add the next term to get another sum, and on, and on: 1/2 + 1/4 = 3/4, then 3/4 + 1/8 = 7/8, then 7/8 + 1/16 = 15/16, then 15/16 + 1/32 = 31/32. By simple inspection, you can see that the sum is increasing, but is "creeping up" by smaller and smaller increments. You can see where the series is going. It will go to infinity, but each term is getting smaller and smaller. Thinking about it, the terms will approach zero as they get infinitely large. And each term will bring the series closer to summing to 1. At infinity, the sum will be 1.
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How can an infinite series have a finite sum? - Answers
It is possible for an infinite series to have a finite sum if it is of a "special" kind. Consider the infinite series of fractions that is 1/2, 1/4, 1/8, 1/16,...1/n. You can see that each term is 1 over twice the denominator of the previous term. (We might have written the last term as 1/2n where n is the number of the term.) In this series there are an infinite number of elements, but the sum will be 1. In the case of the series, 1/2 + 1/4 + 1/8 + 1/16 +... + 1/n, the sum has a limit of 1, and 1 will be the sum of the terms at infinity. Let's do the math. Add the first two terms, get a sum, than add the next term to get another sum, then add the next term to get another sum, and on, and on: 1/2 + 1/4 = 3/4, then 3/4 + 1/8 = 7/8, then 7/8 + 1/16 = 15/16, then 15/16 + 1/32 = 31/32. By simple inspection, you can see that the sum is increasing, but is "creeping up" by smaller and smaller increments. You can see where the series is going. It will go to infinity, but each term is getting smaller and smaller. Thinking about it, the terms will approach zero as they get infinitely large. And each term will bring the series closer to summing to 1. At infinity, the sum will be 1.
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How can an infinite series have a finite sum? - Answers
It is possible for an infinite series to have a finite sum if it is of a "special" kind. Consider the infinite series of fractions that is 1/2, 1/4, 1/8, 1/16,...1/n. You can see that each term is 1 over twice the denominator of the previous term. (We might have written the last term as 1/2n where n is the number of the term.) In this series there are an infinite number of elements, but the sum will be 1. In the case of the series, 1/2 + 1/4 + 1/8 + 1/16 +... + 1/n, the sum has a limit of 1, and 1 will be the sum of the terms at infinity. Let's do the math. Add the first two terms, get a sum, than add the next term to get another sum, then add the next term to get another sum, and on, and on: 1/2 + 1/4 = 3/4, then 3/4 + 1/8 = 7/8, then 7/8 + 1/16 = 15/16, then 15/16 + 1/32 = 31/32. By simple inspection, you can see that the sum is increasing, but is "creeping up" by smaller and smaller increments. You can see where the series is going. It will go to infinity, but each term is getting smaller and smaller. Thinking about it, the terms will approach zero as they get infinitely large. And each term will bring the series closer to summing to 1. At infinity, the sum will be 1.
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- og:descriptionIt is possible for an infinite series to have a finite sum if it is of a "special" kind. Consider the infinite series of fractions that is 1/2, 1/4, 1/8, 1/16,...1/n. You can see that each term is 1 over twice the denominator of the previous term. (We might have written the last term as 1/2n where n is the number of the term.) In this series there are an infinite number of elements, but the sum will be 1. In the case of the series, 1/2 + 1/4 + 1/8 + 1/16 +... + 1/n, the sum has a limit of 1, and 1 will be the sum of the terms at infinity. Let's do the math. Add the first two terms, get a sum, than add the next term to get another sum, then add the next term to get another sum, and on, and on: 1/2 + 1/4 = 3/4, then 3/4 + 1/8 = 7/8, then 7/8 + 1/16 = 15/16, then 15/16 + 1/32 = 31/32. By simple inspection, you can see that the sum is increasing, but is "creeping up" by smaller and smaller increments. You can see where the series is going. It will go to infinity, but each term is getting smaller and smaller. Thinking about it, the terms will approach zero as they get infinitely large. And each term will bring the series closer to summing to 1. At infinity, the sum will be 1.
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