# Category Archives: sequences and series

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

## 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

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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 Euler-Mascheroni constant . … Continue reading

## 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

## A Bonus Question on Convergent Series

Occasionally when teaching the sequences and series material in second-semester 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

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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

## Generating Function for the Reciprocals of the Central Binomial Coefficients

In this post we generalize the result from the last post to find the generating function for the reciprocals of the central binomial coefficients.  As we did with that one, we start with the beta integral expression for : . Now, … 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