The harmonic series is very imp...................for even more reasons.....................they all have a numerator of 1......................2 + 2 = 4 ..................2 * 2 = 4......................0 + 0 = 0 ............0 * 0 = 0.............no other place in the number line does that happen.............when u use the same number.........................as in one.............the same one............................of course there is a difference in 0 and 2..........in that using two yields a different number other than itself...............zero never differs..........................zeros straight across..................

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:

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

Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than the sum of the second series. However, the sum of the second series is infinite: