Sum Of The Infinite Geometric Series -3 -3/2 -3/4

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Understanding Infinite Geometric Series

In the realm of mathematics, infinite geometric series represent a fascinating concept where an infinite number of terms, each related to the previous one by a constant ratio, are summed together. These series appear in various mathematical and scientific applications, from calculating compound interest to modeling radioactive decay. Understanding how to determine the sum of such a series is a fundamental skill in calculus and analysis. This article delves into the intricacies of infinite geometric series, explaining the conditions under which they converge, the formula for calculating their sum, and providing a step-by-step solution to the given problem.

To truly grasp the concept, it's essential to first define what a geometric series is. A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio (r). For example, the series 2 + 4 + 8 + 16 + ... is a geometric series with a common ratio of 2, as each term is twice the previous term. When this series extends infinitely, it becomes an infinite geometric series. However, not all infinite geometric series have a finite sum. The convergence of an infinite geometric series depends critically on the value of the common ratio. If the absolute value of the common ratio, |r|, is less than 1, the series converges, meaning its sum approaches a finite value. Conversely, if |r| is greater than or equal to 1, the series diverges, and its sum tends towards infinity or oscillates without approaching a specific value. This distinction is crucial in determining whether we can meaningfully calculate the sum of an infinite geometric series.

The formula for the sum (S) of a convergent infinite geometric series is given by:

S=a1−rS = \frac{a}{1 - r}

where a is the first term of the series and r is the common ratio. This formula is derived from the limit of the partial sums of the geometric series as the number of terms approaches infinity. It elegantly captures the essence of how the terms of a convergent series diminish in magnitude, allowing the sum to approach a finite limit. The condition |r| < 1 is paramount here because, as n (the number of terms) approaches infinity, r^n approaches zero only when |r| is less than 1. If this condition is not met, the series does not converge, and the formula cannot be applied. The formula's simplicity belies its power in solving a wide range of problems involving infinite geometric series. From financial calculations to physics problems, this formula serves as a cornerstone for understanding and quantifying phenomena involving infinite sequences with a constant ratio.

Analyzing the Given Series: $-3 - \frac{3}{2} - \frac{3}{4} - \frac{3}{8} - \frac{3}{16} - \ldots$

To solve the problem at hand, we must first identify the key parameters of the given infinite geometric series: $-3 - \frac{3}{2} - \frac{3}{4} - \frac{3}{8} - \frac{3}{16} - \ldots$. The initial step in determining the sum of any geometric series is to identify the first term (a) and the common ratio (r). The first term, a, is simply the first number in the series, which in this case is -3. The common ratio, r, is the constant value by which each term is multiplied to obtain the next term. To find r, we can divide any term by its preceding term. For instance, dividing the second term (-3/2) by the first term (-3) gives us:

r=−32−3=12r = \frac{-\frac{3}{2}}{-3} = \frac{1}{2}

Similarly, dividing the third term (-3/4) by the second term (-3/2) also yields 1/2, confirming that the common ratio, r, is indeed 1/2. This consistent ratio is a hallmark of geometric series and is crucial for applying the formula for the sum of an infinite geometric series. Once we have identified a and r, the next critical step is to check the convergence condition. As mentioned earlier, an infinite geometric series converges only if the absolute value of the common ratio is less than 1. In mathematical terms, this condition is expressed as |r| < 1. In our case, r = 1/2, so |r| = |1/2| = 1/2, which is indeed less than 1. This confirms that the given series converges, meaning it has a finite sum that we can calculate using the formula.

Having established that the series converges, we can now confidently apply the formula for the sum of an infinite geometric series. This formula, as stated earlier, is:

S=a1−rS = \frac{a}{1 - r}

where S represents the sum of the series, a is the first term, and r is the common ratio. In our specific example, we have already identified that a = -3 and r = 1/2. Plugging these values into the formula, we get:

S=−31−12S = \frac{-3}{1 - \frac{1}{2}}

This expression allows us to directly calculate the sum of the infinite geometric series by performing a simple arithmetic calculation. The denominator, 1 - 1/2, simplifies to 1/2. Thus, the expression becomes:

S=−312S = \frac{-3}{\frac{1}{2}}

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we multiply -3 by the reciprocal of 1/2, which is 2:

S=−3×2S = -3 \times 2

This final calculation gives us the sum of the infinite geometric series, which is -6. This result is a precise and finite value, demonstrating the convergence of the series and the power of the formula in calculating such sums.

Calculating the Sum

With the first term (a = -3) and the common ratio (r = 1/2) identified, and the convergence condition (|r| < 1) satisfied, we can now proceed to calculate the sum of the infinite geometric series. The formula for the sum (S) of an infinite geometric series is:

S=a1−rS = \frac{a}{1 - r}

Substituting the values of a and r into the formula, we get:

S=−31−12S = \frac{-3}{1 - \frac{1}{2}}

First, we simplify the denominator:

1−12=22−12=121 - \frac{1}{2} = \frac{2}{2} - \frac{1}{2} = \frac{1}{2}

Now, substitute this back into the equation:

S=−312S = \frac{-3}{\frac{1}{2}}

To divide by a fraction, we multiply by its reciprocal:

S=−3×21=−3×2S = -3 \times \frac{2}{1} = -3 \times 2

Finally, we calculate the sum:

S=−6S = -6

Therefore, the sum of the infinite geometric series $-3 - \frac{3}{2} - \frac{3}{4} - \frac{3}{8} - \frac{3}{16} - \ldots$ is -6. This result demonstrates how the terms of the series, though infinitely many, converge to a finite sum due to the diminishing magnitude of each subsequent term. The negative sum reflects the fact that all terms in the series are negative, and their cumulative effect approaches -6 as we consider more and more terms.

Conclusion

In summary, the sum of the infinite geometric series $-3 - \frac{3}{2} - \frac{3}{4} - \frac{3}{8} - \frac{3}{16} - \ldots$ is -6. This result is obtained by carefully analyzing the series to identify the first term (a) and the common ratio (r), verifying that the convergence condition (|r| < 1) is met, and then applying the formula for the sum of an infinite geometric series. The steps involved in this process are crucial for solving similar problems and understanding the behavior of infinite geometric series in general.

The concept of infinite geometric series and their sums is a powerful tool in mathematics and has broad applications in various fields. From calculating the present value of a perpetuity in finance to modeling the decay of radioactive substances in physics, the principles of geometric series are fundamental. The key takeaway is the importance of the common ratio in determining convergence. Only when the absolute value of the common ratio is less than 1 does the series converge to a finite sum, allowing us to use the formula S = a / (1 - r). Understanding this condition and the formula itself is essential for anyone studying calculus, analysis, or related disciplines.

Furthermore, the example provided highlights the importance of careful calculation and attention to detail when working with infinite series. Each step, from identifying the first term and common ratio to verifying convergence and applying the formula, must be performed accurately to arrive at the correct result. The ability to break down a complex problem into smaller, manageable steps is a valuable skill in mathematics and problem-solving in general. As we have demonstrated, even an infinite series can be tamed and its sum precisely determined through a systematic and logical approach. This underscores the elegance and power of mathematical tools in unraveling the complexities of the infinite.