Shall I compare thee to a harmonic series?
Thou art more lovely and more temperate.
Rough winds do shake the n'th terms to zeros
And its divergence hath all too slow a rate.
Sometime to infinity this sequence shines,
And arbitrarily large are his partials summ'd.
And every term from term sometime declines
By chance or nature's changing course, untrimm'd.
But thy eternal series shall not fade,
Nor lose possession of that fair limit thou ow'st,
Nor shall Fourier brag that thou wander'st in his shade,
When in infinite sums to Time, thou grow'st.
So long as Cauchy can breathe and Gauss can see,
So long converges this, and this gives life to thee.
Notes
My past few weeks have been filled with Shakespeare and I feel compelled to translate one of his most famous sonnets into mathematics. The difficult part is in maintaining the rhyme and syllable scheme.
The harmonic series is an infinite series which represents the harmonic tones in music. It is often considered “beautiful” in mathematics and is popularly used by architects to define portions and establish harmonic relationships between different spaces.
As the number of terms in a harmonic series becomes infinite, the limit of its ‘th term approaches zero. However, despite that fact, the harmonic series is still a divergent series – albeit at an extremely slow rate. The sums of its partial terms become infinitely large even as the value of the next term decreases over time by nature of the series.
Convergent infinite series is used as a representation of the love’s beauty. Like an inifnite series, their beauty shall never fade. As is required of convergence, the series never loses its limit.
The Fourier transform is a powerful tool used in harmonic analysis to determine divergence. Gauss and Cauchy also discovered many of the tests used today to determine convergence.
As long as these verses are read, so long lives my love.