nth term test | IA Groups

Ponder...

nth term test

If converges then : a necessary, but not sufficient, condition for convergence of a series.

Proposition 1.27 ( Term Test)

A necessary condition for to converge is that as .

[i.e., if does not converge to , then diverges.]

Proof

If converges, the partial sums converge to some , so is Cauchy. Then

The converse fails: the harmonic series has terms tending to , yet diverges, because its partial sums satisfy

so is not Cauchy.