Question

if a series is absolutely convergent, then it is also convergent?

Answer 1

hi! it's false right?

Answer 2

It's true...

Answer 3

hmm
what if

Answer 4

{An} and {Bn} are convergent, then {An}+{Bn} must be convergent?

Answer 5

this is also true i guess

Answer 6

Yup :)

Answer 7

what if {An} and {Bn} both divergent, then {An} - {Bn} must be divergent?

Answer 8

It's true...absolute convergence implies the convergence of the series as the terms go to infinity.

Answer 9

Of course not, what if An=Bn ?
then An-Bn = 0 for all n, and is obviously convergent.

Answer 10

\[0 \le Bn \le An and \sum_{?}^{?} (An) converges, then \sum_{?}^{?}(Bn) converges?\]

Answer 11

^that is true. If a bigger series converges, then it makes sense that a smaller series also converges, right?

Answer 12

That's called the comparison test.

Answer 13

\[a sequence {An} converges \to \frac{ 1 }{ 2 } , then \sum_{\infty}^{n=1}(An) must diverge?\]

Answer 14

a sequence An converges to*

Answer 15

For a series to be convergent, then its underlying sequence must converge... to ZERO.

Answer 16

oh i see

Answer 17

And by underlying sequence I believe what he's referring to the sequence of the increments by which each successive term in the sequence of partial sums increases by.

Answer 18

^ So if the sequence which generates the series does not converge to zero (either it converges to something else, or it diverges), then the series must be divergent.

Answer 19

\[a sequence {An} converges \to \frac{ 1 }{ 2 } , then \sum_{\infty}^{n=1}(An) must diverge?\]

Answer 20

so for this one

Answer 21

So for this one, what do you think?

Answer 22

it not true

Answer 23

......if you have a shirt then you have at least one cloth.......T/F..same thing