Exploring the Minimum Infinite Sum: A Fun Algebra Challenge
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Understanding Infinite Sums
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Before diving into the solution, remember that S must be a positive value. I suggest you take a moment to pause this article, get your pen and paper, and attempt the problem yourself. Once you’re ready, continue for the solution!
Solution Overview
Let’s denote the common ratio of the geometric series as r. Given that the second term equals 1, it follows that the first term can be expressed as 1/r. We can start writing the series terms as follows:
In this context, we apply the formula S = a/(1 - r), where a represents the first term, and r is the common ratio. For our scenario, we have a = 1/r, as shown previously.
Simplifying this expression results in:
Notice that since S > 0 and S is equal to 1/(r - r²), we can minimize S by maximizing the expression r - r². This allows us to reframe our problem as one concerning a downward-opening parabola, f(r) = r - r².
The vertex of this function occurs at (0.5, 0.25), indicating that 0.25 is the peak value. You could also arrive at this conclusion through methods such as completing the square or taking the derivative. Therefore, the smallest possible value for S is:
And that concludes our solution.
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Chapter 2: Video Insights on Infinite Sums
In this chapter, we will explore two insightful videos to deepen our understanding of infinite sums.
The first video titled "Infinite Series: Minimum Number of Terms for a Given Accuracy" provides an excellent explanation of the topic.
The second video, "13.2 The Definition of Infinite Sum," further elaborates on this fascinating subject.