Thread: 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.

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: does not converge

but its term does approach 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: .

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

False

The oscillating series: . 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.

I'll let you think about this one . . .

Originally Posted by TreeMoney

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

Yes, ever convergent sequence (series is a type of sequence) is bounded.