Question

Suppose that a series sigma an has positive terms and its partial sums sn satisfy the inequality sn <= 1000 for all n. Explain why the sum of an must be convergent.

Answer 1

Since all the terms are positive, the partial sums sequence is increasing.
This is because adding a positive term always gives you a bigger number

Answer 2

Ok.

Answer 3

Also \(1000\) is an upper bound. So by monotone convergence theorem, the partial sums sequence converges.

"The partial sums sequence converges" is same as saying "the series converges"

Answer 4

http://en.wikipedia.org/wiki/Monotone_convergence_theorem

Answer 5

Wait how is saying that when the sequence converges and so the series converges the same?

Answer 6

How does that work in other words.

Answer 7

sequence of "partial sums" converges

is same as saying

the series converges

Answer 8

How? Could you give me an example please? :)

Answer 9

lets look at a quick example maybe

Answer 10

Hahaha, same idea!

Answer 11

pick your favorite converging series

Answer 12

Whoops. Oh yeah.

Answer 13

Hahaha..ha.h.a.ha.

Answer 14

pick some easy converging geometric series maybe

Answer 15

1/n(n+1)?

Answer 16

1/n(n+1) is NOT a series

Answer 17

that looks like an algebraic expression to me

Answer 18

a series must have "+" symbols or a \(\sum\) symbol

Answer 19

5, -10/3, 20/9, -40/27 added together

Answer 20

can you pick something simpler such that all terms are positive

Answer 21

5, 10/3, 20/9, 40/27...? :)

Answer 22

try picking a more simpler one

Answer 23

how about

1/1, 1/2, 1/4, 1/8, 1/16, 1/32, ...

?

Answer 24

Sure!

Answer 25

thnks

Answer 26

is that a sequence or series ?

Answer 27

sequence.

Answer 28

good, whats the first partial sum \(S(1)\) ?

Answer 29

1

Answer 30

whats the second partial sum \(S(2)\) ?

Answer 31

S(2) = 3/2
S(3) = 7/4

Answer 32

how about the first 6 parital sums ?

\(S(1) = ?\)
\(S(2) = ?\)
\(S(3) = ?\)
\(S(4) = ?\)
\(S(5) = ?\)
\(S(6) = ?\)