Happy New Year!

Now that you’ve caught up on your sleeping and filled yourself with too much food, you probably have a hankerin’ for a math problem. Here is one we particularly like because it only uses Calc III sequence and series stuff:

Everybody knows that the harmonic series

diverges (since it’s partial sums grow without bound, it can’t converge to any fixed number). One amusing consequence of this is the fact that you can build a bridge from New York to London by simply stacking playing cards on top of each other.

Now let’s consider a closely related series. Pick your favorite digit between 2 and 9. We’ll pick 2. Make a new series by taking the harmonic series and deleting all terms which have your fixed digit in it one or more times. So we would delete 1/2, 1/12, 1/20, 1/21, 1/22, …., 1/8591721, etc. Of course, the majority of the terms of the harmonic series will remain undeleted. This new series is the one we want to think about.

Question: Does this new series converge?

Question: Does whether or not it converge depend on which digit you delete?

Question: If it converges, say to the number S, you know that it can be very difficult to compute what S is for most series. Instead, can you at least give a number which you know is larger than S?

Solutions are welcome in the comments. Hint: The answer to these questions only require Calc III mathematics. We’ll post the solution in a few days or, if you’re impatient, you can google the solution without too much difficulty.

Below is an interesting related question:

On the other hand, we all know that the series

converges since it’s a p-series with . In this case we actually know (thanks to Euler) the series converges to . Here is an amusing question: Notice that . If you think of the above sum as the areas of squares of side length 1/2, 1/3, 1/4, etc., can you tile a square of side length 5/6 with one square of each of these sizes without overlapping the squares?

The answer is yes. This pictures shows how to do it (in the picture n stands for the square of side length 1/n):

Tiling a Square thanks to Erich Friedman

You can find more details here.

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