Harmonic Series Investigation: An Investigation Of The Harmonic Series

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Harmonic Series Investigation
This investigation will examine some aspects of Harmonic Series’. The name harmonic series come from overtones or harmonics in music. This is because the wavelengths in the overtones on a vibrating string are ½, ⅓. ¼ etc. just like the terms in the harmonic series. However, this investigation will mainly focus on testing for convergence or divergence on the harmonic series, as well as other variations of the series rather than the physics behind the numbers. The numbers in the harmonic series have been used for many architectural designs especially during the Baroque period. Harmonic series’ have some interesting properties which I shall i explore throughout this paper.
The Harmonic series can be represented by n=11n= 1 + 12+ 13...1
First I will prove the the Harmonic Series is divergent. The harmonic series seems as if it would converge because each of its terms approach zero, but it is actually divergent. I will use the first few terms of the harmonic series and compare it to the first few terms of another divergent series. The series I compare the harmonic series to was created because it diverges and the sum of each fraction with the same denominator is equal to ½. The divergent series will be used as a known divergent series to be compared to in each of the divergence tests throughout this paper.
Harmonic Series - 1+12+13+14+15+16+17 +18...
Divergent Series - 1+12+14+14+18+18+18+18…
The second series can be proven to diverge by grouping the terms that contain the same denominator ie. ¼ and ¼. The sum of the terms in each of these groups is equal to ½. This combination makes the series appear as this : 1+(12)+(12)+(12)+(12)…
The series above is divergent because the halfs add up infinitely ther...

... middle of paper ...

...uared ie.x=11n2= 1+14+19+116…
This series is convergent using the p-series test because the value of p=2 and when p>1, the series is convergent.
In this portfolio, I have investigated information about the harmonic series and then some variations of harmonic series’. To summarize, I have concluded that the harmonic series in divergent through comparing it to another known divergent series and through the improper integral test. I have proved divergence for a few other invented series through the same tests. I also showed the alternating harmonic series and proved that it converged to ln2 using the taylor series. I then proved that any series in the form n=11an+b is divergent and its sum is infinite. Finally in this investigation i modified the harmonic series by putting an exponent in the denominator and using the p-series test to prove convergence or divergence.

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