AP Calculus BC · Unit 10 Infinite Sequences & Series
Learning Objectives
Infinite Sums: Does adding infinitely many numbers result in infinity? Not if the terms shrink fast enough to zero!
The Ratio Test: Checks how fast consecutive terms shrink: $L = \lim |a_{n+1} / a_n|$. If $L < 1$, it converges wildly fast (Absolute Convergence).
Alternating Series: If terms alternate $+,-,+,-$, it converges purely because the additions and subtractions cancel each other out, locking the sum in a trapped boundary!
Tags
Infinite SeriesRatio TestASTp-Series
PARTIAL SUM EVOLUTION
$S_{N} = \sum_{n=1}^{N}$ a_n
Term Magnitude $a_n$
Partial Sum $S_n$
Select Infinite Series
Interactive Accumulation
Terms Added $N$ = 10
Convergence Test Results
CURRENT SERIES STATE N=10
Adding $a_{N} =$0.00
Current Sum $S_{N} =$0.00
THEORETICAL OUTCOME $\infty$
Test Output:-
DIVERGES
The Paradox: The terms $1/n$ approach 0. But they don't approach 0 fast enough! Try sliding N. The sum looks like it's leveling off, but it will cross $100$, then $1000$... it crawls to infinity.