29 lines
1.5 KiB
Text
29 lines
1.5 KiB
Text
In mathematics, the '''n-th''' harmonic number is the sum of the reciprocals of the first '''n''' natural numbers:
|
||
|
||
<!-- mathml not working
|
||
<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><mi>k</mi></mfrac></math>
|
||
-->
|
||
<big>'''H<sub>''n''</sub> = 1 + 1/2 + 1/3 + ... + 1/n'''</big>
|
||
|
||
The series of harmonic numbers thus obtained is often loosely referred to as the harmonic series.
|
||
|
||
Harmonic numbers are closely related to the [[wp:Riemann_zeta_function|Riemann zeta function]], and roughly approximate the [[wp:Natural_logarithm|natural logarithm function]]; differing by <big>γ</big> (lowercase Gamma), the [[wp:Euler–Mascheroni_constant|Euler–Mascheroni constant]].
|
||
|
||
The harmonic series is divergent, albeit quite slowly, and grows toward infinity.
|
||
|
||
|
||
;Task
|
||
* Write a function (routine, procedure, whatever it may be called in your language) to generate harmonic numbers.
|
||
* Use that procedure to show the values of the first 20 harmonic numbers.
|
||
* Find and show the position in the series of the first value greater than the integers 1 through 5
|
||
|
||
|
||
;Stretch
|
||
* Find and show the position in the series of the first value greater than the integers 6 through 10
|
||
|
||
|
||
;Related
|
||
* [[Egyptian fractions]]
|
||
|
||
<br>
|
||
|