Also, we are now twittering at MathFactor; each of the authors has an account of his own; mine is CGoodmanStrauss. You can tag solutions and comments with #mathfactor. See you there!

Check out these great gifts!

• Zome is an incredibly powerful construction system!
• the great puzzles of Puzzellation (available at Barnes and Nobles)
• The terrific puzzle computer game DROD
• The Magic Mirror Image Coloring Book
• The Riddles of the Sphinx by David J Bodycombe, an amazing compendium of puzzles, of hundreds of kinds, at all levels of difficulty, with historical essays to boot!
• Which leads us to Nikoli, the great Japanese puzzle co! (Rules can be found here)
• The Princeton Companion to Mathematics is a landmark classic. A must-have for every serious student, researcher or amateur.
• How Round is Your Circle just one of the many fantastic titles out on Princeton University Press
• AK Peters is another fantastic press, with a wide range of interesting math and CS titles, including, ahem, the Symmetries of Things.
• Binary Arts/ThinkFun is another source of great puzzles!
• And the authors Martin Gardner and Ivan Moscovitch are always fantastic!

Hope this helps and have fun!! Let us know how it works out!

Happy Holidays from the Math Factor!

1 + 1/2 + 1/3 + 1/4 + . . .
diverges to infinity, growing as large as you please!

If you try adding terms up on a calculator, this scarcely seems possible! By the time you have added the hundredth term, you will have a reached only a whopping 5.187… (and each new term will be less than .01).

After adding up a million terms, you will have made it only to about 14.39272672… — and each new term will be less than .000001. Does the series really diverge?

The eighteenth century mathematician Jacob Bernoulli gave a very elegant proof that it does:

1/2 is at least 1/2

1/3 + 1/4 is at least 1/4 + 1/4 = 1/2

1/5 + 1/6 + 1/7 + 1/8 is at least 1/8 + 1/8 + 1/8 + 1/8 = 1/2

1/9 + . . . + 1/16 is at least 8 x 1/16 = 1/2

etc. So the result of adding up the first 2ⁿ terms 1/2 + 1/3 + . . . + 1/2ⁿ is at least n/2, and in particular, can be as large as we please.

But this does take a long time to get anywhere. To add up to, say, 100, Bernoulli’s proof shows us that 2¹⁹⁸ (about 4×10⁵⁹) terms will suffice. But maybe this is more than we actually need.

A basic fact from calculus is that the area under the curve y = 1/x, from x = 1 to x = N is exactly ln N.

Now the area of a box 1 unit wide and 1/n units tall is 1/n, and boxes of width 1 and heights 1, 1/2, 1/3, . . . altogether have area 1 + 1/2 + 1/3 . . .

Here we see that these boxes can be arranged to show that

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 > ln 8

Shifting the boxes over, we see that

1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 < ln 8

This gives us a much better bound on the harmonic series. Generally, we have that

1 + 1/2 + . . . + 1/n is between ln (n+1) and (ln n) + 1.

So to be sure that the series sums to at least 100, we can be sure that e^100-1 (about 2.7×10⁴³) terms will suffice!

So the sum of the first million terms is about (ln 1,000,000) + γ, and if we want to sum to 100, we need to have n such that ln n + γ is greater than 100; in other words, e ^{(100 – γ)} (about 1.5×10⁴³) terms will do.

The series Σ 1/(n ln n) diverges even more slowly still, taking about e^e^n terms to sum to n (!!) The series Σ 1/(n (ln n) (ln ln n)) takes e^e^e^n terms to sum to n. Etc!!

Dana Richards, editor of The Colossal Book of Short Puzzles and Problems discusses the amazing Martin Gardner and his legacy!

Faster than an exponential! More powerful than double factorials!! The Busy Beaver Function tops anything that could ever be computed– and we mean ever

Those dumb robots can do anything! Anything at all, that any computer can do.

Can the answer to any mathematical question be computed?

We’ve also prepared a more comprehensive, much more subtle discussion here.

Of course we misspoke in the podcast when we said that Goldbach conjectured that every even number is the sum of two primes &emdash; 2 itself is not!