In fact, somewhat counterintuitively, some infinite geometric sequences do have a finite sum. In fact, as □ approaches infinity for this sequence, the sum of the terms, □ , will also approach infinity. We might infer, then, that if we were to calculate the sum of a large number of terms, our result would be particularly large. We notice that as the term number, □, increases, the value of the term itself, □ , grows exponentially larger. Now, let’s go back to our earlier example of a geometric sequence: 1, 3, 9, 2 7, 8 1, …. Īlternatively, it can be also given by □ = □ □. The common ratio, □, of a geometric sequence whose □th term is □ is given by, □ = □ □. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License In each case, find the minimum value of N N such that the N th N th partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Use this expression to compute the maximum overhang (the position of the edge of the top block over the edge of the bottom block.) See the following figure.Įach of the following infinite series converges to the given multiple of π π or 1 / π. This implies that ( N − 1 ) x = ( 1 2 − x ) ( N − 1 ) x = ( 1 2 − x ) or x = 1 / ( 2 N ). Let x x denote the position of the edge of the bottom block, and think of its position as relative to the center of the next-to-bottom block. ![]() Archimedes’ law of the lever implies that the stack of N N blocks is stable as long as the center of mass of the top ( N − 1 ) ( N − 1 ) blocks lies at the edge of the bottom block. Suppose that N N equal uniform rectangular blocks are stacked one on top of the other, allowing for some overhang. Show that ln ( k + 1 ) − ln k Prove that for k ≥ 1, k ≥ 1, 0 < T k − γ ≤ 1 / k 0 < T k − γ ≤ 1 / k by using the following steps. ![]()
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