1. ## Series

I have to answer the following true or false questions. But also be able to explain why. I think I have the first 2 right saying they are both true. But I can't really explain why and I don't know about the last 2.

True or False

a) If a series does not converge, then its nth term does not approach 0 as n approaches infinity.

b) If the nth term of a series does not approach 0 as n approaches infinity, then the series diverges.

c) If all partial sums of a series are less than some constant L, then the series converges.

d) If a series converges, then there is a constant L such that all of its partial sums are less than L.

2. Hello, TreeMoney!

Here are a few of them . . .

True or False

a) If a series does not converge, then its nth term does not approach 0
as n approaches infinity.

False

The Harmonic Series: $\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \hdots$ does not converge

but its $\displaystyle n^{th}$ term does approach 0: .$\displaystyle \lim_{n\to\infty}\frac{1}{n} \:=\:0$

b) If the nth term of a series does not approach 0 as n approaches infinity,
then the series diverges.

True

A necessary condition for convergence is: .$\displaystyle \lim_{n\to\infty} a_n \:=\:0$

c) If all partial sums of a series are less than some constant L,
then the series converges.

False

The oscillating series: .$\displaystyle 1 - 1 + 1 - 1 + \hdots$ has a sum less than 2,
. . but it diverges.

d) If a series converges, then there is a constant L
such that all of its partial sums are less than L.