If ∑an converges, then necessarily lim⁡n→∞ $a_n$n=0.

Think about it:

If you're adding infinitely many numbers and the numbers don't shrink to zero, you keep adding noticeable amounts forever. The total can't settle down.

Series limit limSn asks:

What happens to the running total?

This can be any finite number.

Never conclude convergence from

$lim a_n=0$ alone.

Geometric Series

converges ONLY if $|r|<1$

diverges if $|r|≥1$

Finding the Sum (when it converges)

Sum is $a/(1-r)$

Telescoping Series

• Everything cancels in a chain