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 | Leave a comment

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

Posted in calculus, sequences and series | 6 Comments

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

Posted in binomial coefficients, sequences and series | Leave a comment

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

Posted in binomial coefficients, generating functions, sequences and series | 1 Comment

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 | 1 Comment

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 | 2 Comments

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

Posted in Fibonacci sequence, recurrence relations, sequences and series | Leave a comment