[Verse 1]
Picture functions as infinite polynomials in disguise
Each coefficient tells a story, coefficients never lie
From the center point c, we build our infinite sum
A-sub-n times x minus c to the power n, here we come
[Chorus]
Power series, power series, sigma n equals zero to infinity
A-sub-n times x minus c to the n-th degree
Radius R equals one over limit supremum of a-sub-n to the one over n
Inside the radius absolute convergence, outside divergence begins
At the boundary anything can happen, that's where mysteries live within
[Verse 2]
E to the x has radius infinite, x to the n over n factorial
Sine and cosine stretch forever too, their domains truly global
But one over one minus x stops at one, geometric series bound
Natural log of one plus x the same, where singularities are found
[Chorus]
Power series, power series, sigma n equals zero to infinity
A-sub-n times x minus c to the n-th degree
Radius R equals one over limit supremum of a-sub-n to the one over n
Inside the radius absolute convergence, outside divergence begins
At the boundary anything can happen, that's where mysteries live within
[Bridge]
When absolute value x minus c is less than R
Convergence is guaranteed, both absolutely and uniformly far
When absolute value x minus c exceeds our magic R
The series explodes to infinity, divergence is the star
Complex singularities determine where the radius ends
Even real functions bow to complex analysis, my friends
[Verse 3]
Differentiate and integrate term by term inside
The radius is your playground where calculus can ride
Every smooth function hides an algebraic heart
Power series strip away the mask, reveal the polynomial art
[Chorus]
Power series, power series, sigma n equals zero to infinity
A-sub-n times x minus c to the n-th degree
Radius R equals one over limit supremum of a-sub-n to the one over n
Inside the radius absolute convergence, outside divergence begins
At the boundary anything can happen, that's where mysteries live within
[Outro]
Every function is a polynomial
When you know where to look
Every function is a polynomial
Written in convergence book