[Verse 1]
Start with one, then r, then r squared
Each term shrinking, perfectly paired
The magic happens when r stays small
Less than one in absolute call
Add them up from zero to infinity's door
When will this endless sequence soar?
[Chorus]
One over one minus r
That's where infinite sums are
Sigma k equals zero rising high
r to the k beneath the sky
But only when the ratio's tame
Absolute value less than one's the game
Geometric series, crystal clear
One over one minus r we cheer
[Verse 2]
Half plus quarter plus eighth in line
Point five ratio, the pattern's fine
Converges to exactly two
The formula proves it's always true
But try with r equals three instead
The series explodes, goes off ahead
[Chorus]
One over one minus r
That's where infinite sums are
Sigma k equals zero rising high
r to the k beneath the sky
But only when the ratio's tame
Absolute value less than one's the game
Geometric series, crystal clear
One over one minus r we cheer
[Bridge]
For finite sums when n cuts the chain
One minus r to n plus one domain
Divide by one minus r again
The partial sum formula's refrain
From finance calculations to decimal spots
This series connects so many dots
[Verse 3]
Each term must shrink to reach the prize
Convergence needs that compromise
The ceiling waits, finite and sure
Though terms go on, the sum stays pure
Infinity captured in finite form
The closed expression breaks the norm
[Chorus]
One over one minus r
That's where infinite sums are
Sigma k equals zero rising high
r to the k beneath the sky
But only when the ratio's tame
Absolute value less than one's the game
Geometric series, crystal clear
One over one minus r we cheer
[Outro]
Shrinking terms that find their way
To finite sums that always stay
The prototype of convergence law
Geometric beauty without flaw