Question

Is there a infinite series that its limit goes to 0 however, it is increasing all the time? Or is there a infinite series that limit doesn't equal to 0 however, it is decreasing? (It is about alternate series for Calc 2) P.S. i don't think the (-) count..

Answer 1

well it is like Sn equals to some thing like \[(-1)^{k+1}\times 2/k\] it is decreasing and the limit is to 0 based on the alternate series rules, however i try to find the series with limit goes to 0 however, it is increasing all the time? Or is there a infinite series that limit doesn't equal to 0 however, it is decreasing

Answer 2

Well, for the first on how bout -1/k. It starts at -1, and is constantly increasing to 0.

Answer 3

but i am think not to involve - because that will be kind of meaningless

Answer 4

I am unsure of what you are aking, so it's hard to answer it. Here is the question I think you are asking, and my answer follows.

Question: "Is there an infinite sum serie that, while every one of its terms get closer and closer to zero, the total sum of the terms is still increasing?"

Answer: Yes there is. As EulerSucks suggested suggested,
[ sum( 1/k ) from k=1 to k=infinity] is a good example. Each term gets closer to zero with increasing k, but the overall sum does not converge.

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source:
http://www.wolframalpha.com/input/?i=infinite+sum+&a=*C.infinite+sum-_*Calculator.dflt-&f2=1%2Fx&f=Sum.sumfunction_1%2Fx&f3=1&f=Sum.sumlowerlimit_1&f4=Infinity&x=8&y=5&f=Sum.sumupperlimit_Infinity&a=*FVarOpt.1-_**-.***Sum.sumvariable---.*--

Answer 5

but isn't that 1/k is the term in the series and it is decreasing?

Answer 6

Then I guess I did not understand your question.
Can you restate it?

Answer 7

and what i mean is the term decrease as going on, and the sum of the series does not converge

Answer 8

I thought this is what you where looking for.
\[\sum_{k=1}^{\infty}1/k=\infty\]

If not, then are you looking for a serie that increases, yet converges to zero?

Answer 9

yes

Answer 10

yes what?

Answer 11

i think it is really hard to state the question, hmm, according to the alternating series test,
Suppose that lim ak=0 and 0<ak+1<ak for all k >1 then the alternating series converges, and i am trying to find if there is any series, that meet the limit = 0 however, ak+1>ak or meet ak+1<ak, however limit doesn't equal to 0.

Answer 12

Thanks, I get your question.
and I guess [ 1+ (1/k) ] does not qualify even if ak+1<ak, and limit doesn't equal to 0 ;)
Hum... I don't know if such series exist.
The only thing that I can think of are periodical series like cos(x*PI) where terms alternate around zero.

Maybe someone else will be able to find one.

In the meantime, you can go play on the WolframAlpha website, just type "infinite sum".
Good luck.