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Category Archives: sequences and series
Zeno’s Paradoxes
The ancient Greek philosopher Zeno of Elea is known for proposing several paradoxes related to time, space, motion, and infinity. In this post we’ll focus on one of Zeno’s paradoxes and discuss how some ideas associated with calculus might or … Continue reading
Posted in calculus, infinity, logic, paradoxes, real analysis, sequences and series
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Sum of the Reciprocals of the Binomial Coefficients
In this post we’re going to prove the following identity for the sum of the reciprocals of the numbers in column k of Pascal’s triangle, valid for integers : Identity 1: The standard way to prove Identity 1 is is … Continue reading
Posted in binomial coefficients, sequences and series
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Infinite Series Expressions for Pi, E, Phi, and Gamma
Recently I was trying to find infinite series expressions for some famous mathematical constants, and I thought I would record what I found here. This post considers , Euler’s constant e, the golden ratio , and the EulerMascheroni constant . … Continue reading
Posted in sequences and series
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The Divergence of the Harmonic Series
The fact that the harmonic series, , diverges has been known since the time of Nicole Oresme in the 14th century, but this fact is still somewhat surprising from a numerical standpoint. After all, each successive term only adds a … Continue reading
Posted in harmonic numbers, sequences and series
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A Bonus Question on Convergent Series
Occasionally when teaching the sequences and series material in secondsemester calculus I’ve included the following question as a bonus: Question: Suppose is absolutely convergent. Does that imply anything about the convergence of ? The answer is that converges. I’m going … Continue reading
Posted in calculus, sequences and series
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Alternating Sum of the Reciprocals of the Central Binomial Coefficients
In the last post we proved the generating function for the reciprocals of the central binomial coefficients: In this post we’re going to use this generating function to find the alternating sum of the reciprocals of the central binomial coefficients. … Continue reading
Sum of the Reciprocals of the Central Binomial Coefficients
In this post we prove the formula for the sum of the reciprocals of the central binomial coefficients : . (Of course, the sum of the central binomial coefficients themselves does not converge.) We start with the beta integral, . Replacing n … Continue reading
Posted in binomial coefficients, sequences and series
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A Proof of Dobinski’s Formula for Bell Numbers
Dobinski’s formula entails the following infinite series expression for the nth Bell number : In this post we’ll work through a proof of Dobinski’s formula. We’ll need four formulas: The Maclaurin series for : . The formula for converting normal powers to falling … Continue reading
Posted in Bell numbers, sequences and series, Stirling numbers
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An Explicit Solution to the Fibonacci Recurrence
The Fibonacci sequence is a famous sequence of numbers that starts 1, 1, 2, 3, 5, 8, 13, 21, and continues forever. Each number in the sequence is the sum of the two previous numbers in the sequence. It’s easy to … Continue reading