If sigma Un be series of positive terms and limit Un=0, then sigma n-> infinity Un is:1)Necessarily(N) convergent 2)N divergent

If sigma Un be series of positive terms and limit Un=0, then sigma n-> infinity Un is:
1)Necessarily(N) convergent 2)N divergent 3)Oscillatory

Expert Answers
sciencesolve eNotes educator| Certified Educator

Since the problem provides the information that `lim_(n->oo) u_n = 0` , you need to remember that this information represents the necessary condition of convergence of a infinite series of positive terms such that:

`sum_(n=0)^oo u_n`

You need to remember that the infinite series `sum_(n=0)^oo u_n` is convergent if the partial sums converges.

Hence, using the information provided by the problem, `lim_(n->oo) u_n = 0` , yields that the series `sum_(n=0)^oo u_n ` is necessarily convergent, thus, you need to select the first answer 1) necessarily convergent.

degeneratecircle eNotes educator| Certified Educator

It's possible I'm misinterpreting the problem statement, but I don't believe any of those answers are correct. If the terms approach 0, that tells us nothing about the convergence of the series, since

1+1/2+1/4+1/8+... converges and the terms approach zero,


1+1/2+1/3+1/4+... diverges and the terms also approach zero.