6.5 is dead center .........in the series............2 - 11............................6 + 5 = 11........................the end of the series..................................also.........

where γ is the Euler–Mascheroni constant and ε_k ~ 1/2k which approaches 0 as k goes to infinity. Leonhard Euler proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e.

.

If u take a "top down" approach.........to the harmonic series..................and let the fundamental..........1...........the length of the string of the instrument be 1...........which can also be expressed.........1/1................11.......overlap and interconnection..................then, again.......1/9.....the next number in the h. series......is 1/10.................so if 1/1 is 11...................1/9 is 19 and 1/10..........110............they all begin with 11...........the fundamental............which is the numerator.......in the h series........................................

Rate of divergence[edit]

The harmonic series diverges very slowly. For example, the sum of the first 10⁴³ terms is less than 100.^[9] This is because the partial sums of the series have logarithmic growth. In particular,

${\displaystyle \sum _{n=1}^{k}\,{\frac {1}{n}}\;=\;\ln k+\gamma +\varepsilon _{k}<\ln(k)+1}$

where γ is the Euler–Mascheroni constant and ε_k ~ 1/2k which approaches 0 as k goes to infinity. Leonhard Euler proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e.

${\displaystyle \sum _{p{\text{ prime }}}{\frac {1}{p}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+\cdots =\infty .}$

Partial sums[edit]

The first thirty harmonic numbers[show]

The nth partial sum of the diverging harmonic series,

${\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}},\!}$

is called the nth harmonic number.

The difference between H_n and ln n converges to the Euler–Mascheroni constant. The difference between any two harmonic numbers is never an integer. No harmonic numbers are integers, except for H₁ = 1.^[10]

Related series[edit]

Alternating harmonic series[edit]

The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).

The series

${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\;=\;1\,-\,{\frac {1}{2}}\,+\,{\frac {1}{3}}\,-\,{\frac {1}{4}}\,+\,{\frac {1}{5}}\,-\,\cdots }$

is known as the alternating harmonic series. This series converges by the alternating series test. In particular, the sum is equal to the natural logarithm of 2:

${\displaystyle 1\,-\,{\frac {1}{2}}\,+\,{\frac {1}{3}}\,-\,{\frac {1}{4}}\,+\,{\frac {1}{5}}\,-\,\cdots \;=\;\ln 2.}$

This formula is a special case of the Mercator series, the Taylor series for the natural logarithm. A proof without words that the sum is ln 2 was shown by Matt Hudelson.^[11]

${\displaystyle {\frac {1}{1}}\left({\frac {1}{1}}-{\frac {1}{2}}\right)\,+\,{\frac {1}{2}}\left({\frac {2}{3}}-{\frac {2}{4}}\right)\,+\,{\frac {1}{4}}\left({\frac {4}{5}}-{\frac {4}{6}}+{\frac {4}{7}}-{\frac {4}{8}}\right)+\cdots \;=\;\ln 2.}$