Two sides of one.........a 45/45 degree triangle............................two sides of one......like 21?  The location of the 2nd zero............a triangle........has 180 degrees.............1 + 8 + 0 = 9..............ten 1's.................ten - 9's.............ten - 19's..........by overlapping the pf with the negative integers..............180............is 9 added..........the zero doesn't add......but counts..........zeros......analysis............strange at how it matches up..

A straight line also has 180 degrees..........................ten negative 9s................added to 10 positive 9s.................zeros.....................a straight line........for the hypotenuse..........to gauge the value of a complex number..........its argument........to measure change........the derivative........the straight in the curved................pi.........comparing the curved to a straight.............................






Taking the harmonic series..................it begins with 1...........the only non fraction.............1 can be seen as 1/1............one divided by one is still one...........but using the zero dimension..........and combining the numerator with the denominator............u get 11..................two split in........two ones.........1 + 1 = 2................................a right triangle with two sides of 1.......................has a hyp of the square root of 2.....................1 side 1, the other side one.....................like 11..............

Divergence[edit]

There are several well-known proofs of the divergence of the harmonic series. A few of them are given below.

Comparison test[edit]

One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two:

${\displaystyle {\begin{aligned}&{}1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}+\cdots \\[12pt]>{}&1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots \end{aligned}}}$