Calculus II · Advanced · 5 min
An infinite geometric sum
Shrinking terms can add to a fixed total. Watch the partial sum move toward its target.
WHAT HAPPENS IF…Try r = −0.5. Does the partial sum approach the limit from one side?
Preparing your experiment…
Connect the picture to the ideaSn=1−r1−rn,S∞=1−r1 (∣r∣<1)Subtract rS_n from S_n to cancel all the middle terms. The remaining r^n term disappears only if |r| < 1. At r = 1, the partial sum is n.
A detail that mattersA geometric series with |r| ≥ 1 does not converge, even if its signs alternate.