Question

True or false: An infinite series converges if its sequence of partial sums is bounded and monotone.

Answer 1

monotone sequence is one that is either strictly increasing or decreasing

Answer 2

as for bounded.. I can't think of a way to explain it but something like if the sequence were: -1, 1, -1, 1
the sequence is bounded by [-1,1]

Answer 3

k i would say false, because \[\sum_{0}^{\infty} 1/n\] diverges and thats a monotone series

Answer 4

i didnt give a good example. let me think. I'm 90% sure its false though

Answer 5

hm i think i somewhat understand! the wording can be a mouth full and i get confused ):

Answer 6

k i think i got it. so lets look at my earlier example. thats an infinite series and if we take the sum of some bound, you would have it equaling a number BUT that infinite series still converges as it goes to infinity so its false. anyone else would like to confirm this?

Answer 7

still diverges*

Answer 8

I don't see your example Flop?

Answer 9

I feel like this should be true though as it relates to the Monotone Sequence Theorem.

Answer 10

what does the theorem state?

Answer 11

If a sequence is bounded and monotone, then it converges.

Answer 12

what @kirbykirby said

Answer 13

Now, for series. We know that A series converges if it's sequence of partial sums converges.

Answer 14

i get what your saying and i agree, but the wording in the original question is throwing me off

Answer 15

it just says the partial sums are bounded and monotone. that doesnt necessarily mean it converges, does it?

Answer 16

Well it says the sequence of partial sums is bounded and monotone. By the Monotone Sequence Theorem, a sequence that is bounded and monotone is convergent.

And by definition of convergence of series: A series converges if it's sequence of partial sums converges.

Answer 17

if its sequence*

Answer 18

lets take for example \[\sum_{1}^{\infty} 1/n= 1+ 1/2 +1/3+1/4+.....\] the partial sum of that will converge to some number if we had a bound, but we know that the series diverges as it goes to infinity. what am i thinking wrong?

Answer 19

that example would mean its false, unless im thinking of it all wrong. I dont disagree with the theorem btw, but why is my example wrong then?

Answer 20

does the theorem state that its an infinite series or just a series because that would make a difference

Answer 21

We are looking at the SEQUENCE of partial sums. Not just some random partial sum.

Any partial sum will converge to a number.

Answer 22

It's taking about infinite series.

Answer 23

talking*

Answer 24

alright yea if the sequence converges then the series has to converge. if "bounded and monotone" means convergence then i agree its true

Answer 25

Thank you all for taking the time to answer my question! I believe i got what i was looking for :)

Answer 26

np