Investigation of Harmonic SeriesThis investigation will examine some aspects of harmonic series. The name harmonic series comes from overtones or harmonics in music. This is because the wavelengths of harmonics on a vibrating string are ½, ⅓. ¼ etc. just like the terms of the harmonic series. However, this investigation will focus primarily on tests of convergence or divergence on the harmonic series, as well as other variations of the series rather than the physics behind the numbers. The harmonic series numbers were used for many architectural designs especially during the Baroque period. Harmonic series have some interesting properties that I will explore in this article. The harmonic series can be represented by n=11n= 1 + 12+ 13...1First I will prove that the harmonic series is divergent. The harmonic series appears to converge because each of its terms approaches zero, but in reality it is divergent. I will use the first terms of the harmonic series and compare them with the first 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 equals ½. The divergent series will be used as the known divergent series to compare in each of the divergence tests in this article.Harmonic Series - 1+12+13+14+15+16+17 +18...Divergent Series - 1 +12+ 14+14+18+18+18+18… It is possible to show that the second series diverges by grouping terms containing the same denominator, e.g. ¼ and ¼. The sum of the terms in each of these groups equals ½. This combination makes the series look like this: 1+(12)+(12)+(12)+(12)...The series above is divergent because the halves add to infinity there...... in the center of the 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 investigated information on the harmonic series and therefore some variations of the harmonic series'. To summarize, I concluded that the harmonic series is divergent by comparing it to another known divergent series and through the improper integral test. I have proven divergence for some other invented series through the same tests. I also showed the alternating harmonic series and proved that it converges to ln2 using the Taylor series. I then proved that every series of the form n=11an+b is divergent and its sum is infinite. Finally in this investigation I modified the harmonic series by inserting an exponent in the denominator and using the p-series test to demonstrate convergence or divergence.