Question

what do you understand with this?

Answer 1

Let \[\left\{ a_{n} \right\}:a_{1}, a_{2}, a_{3}, .....,a_{n},...\] be an infinite sequence and let the series \[\sum_{\infty }^{k=1}a_{k}\] converges to S.

Answer 2

Since a_{n}=S_{n}-S_{n-1},\[\lim_{n \rightarrow \infty }S_{n}=S and \lim_{n \rightarrow \infty }S_{n-1}=S \]

Answer 3

\[\lim_{n \rightarrow \infty}a_{n}=\lim_{n \rightarrow \infty}(S_{n}-S_{n-1}=\lim_{n \rightarrow \infty}S_{n}-\lim_{n \rightarrow \infty}S_{n-1}=S-S=0 \]

Answer 4

phew... this is the question..
please help:)

Answer 5

\[a_\infty=?\]

Answer 6

is it 0?

Answer 7

yes that is my understanding.

Answer 8

What exactly does the question ask for?

Answer 9

owh, i found this explanation in one of text book.. but i dont really get what does it mean...can you explain it to me?

Answer 10

it seams like a proof that a sequence converges or something

Answer 11

you have showed the nth term when n is big, is not going to contribute to the sum much

Answer 12

as we add terms Eventually the sum will get close to some value and stay there

Answer 13

Which bit is giving you difficulty?

Answer 14

start here...
Since a_{n}=S_{n}-S_{n-1},lim n→∞ S n =S and lim n→∞ S n−1 =S ..
is it because of previously, we let the series ∑ ∞ k=1 a k converges to S?

Answer 15

What exactly are you asking, Its hard to read the question, with all of unkle's spam respones

Answer 16

@Jack17, actually i dont understand the explanation from one of my text book. is there any simple explanation yet concrete understanding.

Answer 17

What is the question, why don't you upload a picture of the question straight from your textbook.

Answer 18

Also, for questions like this, I wouldn't use this site, its mostly full of 12 year olds in middle school and 30 year olds trying to go back to school.

Answer 19

If you know how to type in tex, a site like mathstackexchange, would be good

Answer 20

i dont have scanner.. sorry..
the question is on 2nd line till 4th line from top....

Answer 21

if you ask a clear question , there will be someone on OS that can help you i am sure

Answer 22

sorry @UnkleRhaukus

Answer 23

Dear all ( @UnkleRhaukus, @Jack17),
here is the explanation from the text book. maybe it is not clear enough.
hope it helps.

Answer 24

Ok , what do you want, that is several lines of equations, showing that $$a_n$$ converges to zero, as $$n$$ tends to infinity

Answer 25

ok.. from that picture. is that the only thing that i need to know or is there any explanition line by line?

Answer 26

Let \(\{ a_n \}:a_1, a_2, a_3, \dots,a_{n},\dots\) be an infinite sequence
and let the series \(\sum\limits^{\infty }_{k=1}a_k\) converge to \(S\)

Since \(a_n=S_n-S_{n-1},\quad\lim\limits_{n\to\infty}S_n=\boxed{S}\) and \(\lim\limits_{n\to\infty}S_{n-1}=\boxed{S}\)

\[\quad\lim\limits_{n\to\infty}a_n=\lim\limits_{n\to\infty}(S_n-S_{n-1})=\lim\limits_{n\to\infty} S_n-\lim\limits_{n\to\infty}S_{n-1}\\
\qquad\qquad\qquad\qquad\qquad\qquad=S-\boxed{S}\\\qquad\qquad\qquad\qquad\qquad\qquad=\boxed{0}\]

so you want to know how they got terms in the boxes?

Answer 27

from: start here
\[\lim\limits_{n\to\infty}S_n=\sum\limits^{\infty }_{k=1}a_k=\boxed{S}\]
and\[\lim\limits_{n\to\infty}S_{n-1}=\sum\limits^{\infty }_{k=1}a_k=\boxed{S}\]
like you said

Answer 28

boy all of this math crap and still no medals or understanding :/

Answer 29

@Jamierox4ev3r..
do you have something to share?

Answer 30

everything that i would have said has already been said, unfortunately

Answer 31

it means that at some point, the stuff you add on will not make a significant difference, compared to the change in n.
example:

sum of 1/n does not converge
sum of 1/n^2 does converge (I like to think it gets there faster)

Answer 32

thanks @zzr0ck3r.. appreciate it:)

Answer 33

Thanks @Jack17 for site suggested

Answer 34

Thanks @UnkleRhaukus