1.3.3—Series convergence

For an infinite series to converge, its terms must tend to zero and the partial sums must approach a finite limit. A geometric series converges when |r|<1; otherwise its partial sums do not settle to a finite sum.

The term test is necessary but not sufficient for every series: terms tending to zero does not by itself prove convergence. Use an appropriate comparison, ratio, integral or known-series test when required.

Σ(1/2)^n converges because the ratio has magnitude below one. Σ1/n has terms tending to zero but still diverges, so checking only the terms is insufficient.

“The terms get small” is not a complete convergence argument, and a finite partial sum is not the value of an infinite series.