SL 1.8—Infinite geometric series

Test convergence before summing an infinite geometric series.

An infinite geometric series has a finite sum only when |r|<1; then S∞=a₁/(1−r).

Worked example
3+1.5+0.75+… converges because r=0.5, giving S∞=6.

Worked example
Does 4+8+16+… have a finite sum? no; |r|=2≥1.

Common boundary
A pattern continuing forever is not automatically summable.