There is a simple and ingenious demonstration of the divergence of the harmonic series (which we talked about last week in relation to the cantilevered stack of books), that is, that the sum 1 + 1/2 + 1/3 + 1 / 4 + 1/5… grows indefinitely as we increase the number of addends. Growth is very slow (it takes septillion addends to reach 100), but undefined.

Effectively:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8… is obviously greater than

1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8… = 1 + 1/2 + 1/2 + 1/2…

And since the second series grows indefinitely, the first, whose corresponding terms are all greater than or equal, also. This demonstration is due to the great medieval mathematician, physicist and astronomer Nicolás de Oresme, who anticipated Copernicus by two centuries by stating that it is the Earth that moves, and not the Sun and the stars. He did it with enough discretion to escape the stake, and for that the credit went to Copernicus and Galileo.

As for the hundredth triangular number, or what is the same, the sum of the first 100 natural numbers, little Gauss found it in a few seconds when he realized that 1 + 100 = 2 + 99 = 3 + 98 = 4 + 97… = 101, so this sum is that of 50 pairs of numbers that add up to 101, that is, 50 x 101 = 5050.

Triangular numbers are so named because they can be represented as sets of points arranged in such a way that they form an equilateral triangle, in which the points on each side indicate the order in the sequence of the triangular (in which 1 is included as the first term ). The Pythagoreans called *tetraktys* to the representation of 10 as an equilateral triangle of points; a configuration that is very familiar to us, since it is equivalent to the usual arrangement of bowling pins in the *bowling*.

Note that 1 + 3 = 4 = 2², 3 + 6 = 9 = 3², 6 + 10 = 16 = 4²… Will the sum of two consecutive triangular numbers always be a perfect square?

**Square numbers**

But why do we call the power 2 of a number the square of that number? Well, for the same reason that we call the numbers we have just seen triangular: because perfect squares or square numbers can be represented as sets of points arranged in such a way that they make up a square. Since 1 is a perfect square, since 1² = 1, in this case its inclusion as the first term in the sequence *goes soi*, as the French say.

The square numbers are, therefore, 1, 4, 9, 16, 25, 36 …

Obviously, the nth square number is n², and n² is equal to the sum of the first n odd numbers:

2² = 1 + 3

3² = 1 + 3 + 5

4² = 1 + 3 + 5 + 7

5² = 1 + 3 + 5 + 7 + 9

Why is this the case for any square number?

A perfect square can end in 0, 1, 4, 5, 6, and 9, but not 2, 3, 7, or 8. This is easily verified by simply squaring all ten digits, since the square of a number it ends up the same as the square of its last figure.

Another property of perfect squares is that they always have an odd number of divisors. Why?

**Carlo Frabetti ***is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn mathematics’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.*

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