Question

Prove or give a counter example:

Answer 1

If \[\sum_{n=1}^{\infty} a_n\] and \[\sum_{n=1}^{\infty} b_n\] are both convergent series with positive terms, then \[\sum_{n=1}^{\infty} b_n * a_n\] is also convergent.

Answer 2

I think the property: If the series of a_n is convergent then lim as n approaches infinity of a_n =0 will help.

Answer 3

I'm stuck on where to go next and it would be terrific if someone could help :)

Answer 4

My teacher gave me a hint and said when lim n→∞ bn = 0 it means that for large enough n, say n > N for some large N, bn ≤ 1? I don't understand this..

Answer 5

Ahh nice that will be a good start

Answer 6

I'm not sure that I understand it though. how does b_n >= 1?

Answer 7

\[\sum\limits_{n=1}^{\infty}b_n\]
if this series converges, then the terms \(b_n\) approach \(0\) as \(n\) gets large, yes ?

Answer 8

yes

Answer 9

that means there exists some index position \(N\), after which all the terms \(b_n\) are less than \(1\), yes ?

Answer 10

Yes

Answer 11

thats what the hint says

Answer 12

Okay... so can we split the series in that sort then?

Answer 13

Yes

Answer 14

\(b_n\le 1 \implies a_nb_n\le a_n\)

we may use comparison test

Answer 15

"Note that since the series of b_n from 1 to infinity is convergent, the lim of b_n as n approaches infinity =0. This means that if n is large enough there exists some index point N after all the terms in the sequence b_n are less than 1.

Answer 16

Would that be a valid statement to write on my homework??

Answer 17

Why do we say n> N?

Answer 18

lets consider a quick example

Answer 19

Awesome! thank you :)

Answer 20

let \(b_n=\frac{10}{n^2}\)
can you plot first few terms of \(\sum\limits_{n=1}^{\infty}b_n\) on number line?

Answer 21

sure thang

Answer 22

right to left, a_1, a_2, a_3, a_4, and a_5