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Chapter 10: Problem 73

Determine whether the infinite geometric series has a sum. If so, find the sum. $$\sum_{k=0}^{\infty} 2(0.5)^{k}$$

Short Answer

Expert verified
Yes, the given series has a sum, and it's 4.

Step by step solution

01

Assessing the validity of the geometric series

First, we need to verify if the common ratio \( r = 0.5 \) is between -1 and 1 (excluding -1 and 1). That is, \( -1 < r < 1 \). Here \( 0.5 > -1 \) and \( 0.5 < 1 \), thus the geometric series is valid and we can proceed with finding its sum.
02

Using the formula to calculate the sum

Once we've validated the geometric series, we apply the formula for the sum of an infinite geometric series \( a/(1 - r) \). Plugging \( a = 2 \) and \( r = 0.5 \) into the formula, we get \( 2/(1 - 0.5) = 4 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Common Ratio
In any geometric series, the common ratio, denoted by \( r \), is the factor between consecutive terms. It's a way to see how the series grows or shrinks with each step. For instance, if we have a term of 2 and the next is 1, the common ratio \( r \) would be \(1/2\), meaning each term is half of the previous one.
  • If \( |r| \) is greater than 1, each term gets larger, and the series possibly stretches to infinity.
  • If \( |r| \) equals 1, the terms stay constant, leading to a non-converging series.
  • If \( |r| \) is less than 1, it converges as terms get smaller and eventually tend towards zero.
In our exercise, the common ratio of our series is 0.5, clearly highlighting a shrinking pattern, as \(0.5 < 1\). This condition lets us proceed with exploring the sum for the infinite series.
Applying the Series Sum Formula
For infinite geometric series, finding the sum implies that the series reaches a limit as it progresses towards infinity. The Series Sum Formula helps in calculating this limit:\[S = \frac{a}{1 - r}\]Where:
  • \( a \) is the first term of the series
  • \( r \) is the common ratio (|r| < 1 for the series to converge)
In our given problem, the first term \( a = 2 \), and the common ratio \( r = 0.5 \). Applying these values to the formula, we find:\[S = \frac{2}{1 - 0.5} = 4\]Thus, the infinite geometric series converges to the sum of 4. This formula provides a way to calculate what infinitely many terms would sum to without manually summing them.
Convergence Criteria in Geometric Series
Convergence is critical in understanding whether an infinite series has a finite sum. For geometric series, the key to convergence is the common ratio \( r \). Specifically, a geometric series converges if the absolute value of \( r \) is less than 1:\[-1 < r < 1\]When \( |r| < 1 \), the terms in the series get progressively smaller, affecting the sum's behavior:
  • The terms approach zero as the series progresses infinitely.
  • This allows the sum to reach a "limit," a finite value.
In our exercise, the common ratio is 0.5, which meets the convergence criteria. This guarantees that despite having infinitely many terms, we arrive at a finite sum, exemplified by our calculated result of 4.

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Most popular questions from this chapter

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