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Category Archives: sequences and series
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