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#1 2013-07-13 00:52:19

rete
Member
Registered: 2013-07-13
Posts: 12

limits of a sum

Hello everybody i  have a couple  of problems
most of them sound like this : lim when n tends to infinity ,( for sum of k=1 to n for [1/(n+k*)])

this k* is important bcoz it keeps changing form, is either, k^2, or sqrt(of k), or sqrt(k-1), or k, etc ...
but the basic problem is how to do it in normal mode

it must be a way to solve this problem but i just have no idea were to start,
if you see a solution, and have it in any detail, it would be much appreciated

also if you have some idea on the following:

f(x)= e^x+nx+3
solve for f(0)(wich i can, easy , just plug in 0 for x and get the answer)
f^(-1) of (3)=
basically sates in the book find f(3) and f^(-1) (3)
i will try and type in latex
any suggestion, comment, idea on any of these would me much apreciated

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#2 2013-07-13 01:35:13

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

Hi;

this k* is important bcoz it keeps changing form, is either, k^2, or sqrt(of k), or sqrt(k-1), or k, etc ...

I am not following you. Please explain what k* means a little more.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#3 2013-07-13 01:53:53

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

it just means that there are 7 problems and that they are all exacly the same exept for the k* part
a is just k
b is k^2
c is sqrt(k)
d is sqrt(k^2+1)
e is sqrt(k^2-1)
f is (k+1)(k+2)..(k+n)
g is (k)!*n!

so if you would ask what c is at the example :
lim when n tends to infinity of (sum from k=1 to n for(1/(n+sqrt(k))

Last edited by rete (2013-07-13 01:55:01)

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#4 2013-07-13 03:34:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

a) log(2)

b) 0

c) 1

d) log(2)

e) log(2)

f) 0

g) 0


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#5 2013-07-13 03:37:15

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

I am not going to argue since i've got no idea on what or how you managed to get any answer, congratulations, but how did you do it?

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#6 2013-07-13 03:37:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

Just playing with the Euler Mclaurin summation formula and series acceleration techniques.

Are these homework problems, or are they out of a book?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#7 2013-07-13 04:15:43

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

they are problems for the summer vacation, for us the nerds, omg, me a nerd, now i have done it, anyways it is just 3 pieces of paper with problems on them, but since i have no idea on how to solve these specific problems, and neither do my colleagues kno were to start at least, I had the idea to ask, found easy ways to solve some of the problems in this sheets, but for these ones, not as much as a hint, (re-looking Euler Mclaurin summation atm,) stil not convinced that this is the way, but since you got some results, might just be worth checking, also finding and just typing results on the paper is of no use, if we get this on a test next year, who will explain this again, we need to understand what we are doing if we are to help others so we get passing grades, and you got 0 and 1 sses, all we got were quote"say infinity, n tends to infinity so infinity", sorry for long post and boring math trolling, but seriously what or how did you get those answers,
later edit:you have a talent or someone showed this type of problems before and now you are just teaching it, anyways, many thx for the reply, congratulations on finding anything different from infinite and if you  have the patience please share the method you used

Last edited by rete (2013-07-13 04:20:19)

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#8 2013-07-13 04:31:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

have the patience please share the method you used

The Euler Mclaurin requires a CAS as the differentiations and Integration are tedious. Series acceleration methods require a computer. They are numerical techniques.

I provided those answers as a signpost. The first step to a full solution is often having the answer first.

Since I do not know where you are at in your math development because you have provided no work it is difficult to look up any possible analytical techniques that might work.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#9 2013-07-13 04:45:13

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

finished 11 grade, but my teacher never bothers to tech in order, we have done some integrals so we can do some problems for the BAC (our final exams in 12th grade) indefinite, definite, ... , matrices , as long as you do NOT use a calculator any method of pen and paper is good method of pen and paper,
later edit: as long as you provide the formula you use and i get the same result for just 1 example, i should get something for other 6 as well, just ask if i kno and i will try to answer to my best of my ability what i understand from what you stated
(I am seriously freaking out here since you got the results so fast and actually might now the generic formulas witch i could apply)

Last edited by rete (2013-07-13 04:51:24)

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#10 2013-07-13 04:52:46

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

So you have not had the summation calculus ( indefinite or definite sums ). In analysis they may ask whether a series converges or not but only in numerical analysis does anyone ask what it sums to.

One of those sums involves the polygamma function, so how can your teacher expect you to be able to sum it at only an 11th grade level?

The only reason I did it quickly is because over the years I have written routines to implement the numerical rules on many types of series.

To give you an idea of how complicated numerical work like this is please read this:

http://www.math.utexas.edu/~villegas/pu … -accel.pdf

If you can understand some of that I will do one using the some of those ideas and do the the best I can to explain it.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#11 2013-07-13 05:38:27

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

my apologies for bad English, basically you are asking me if i kno how much is lim when x tends to infinity of 3x^3/(x^3+etc), just some simple math 3/1=3(no problem we got this
1^ infinit power=(we got this ) a function tends to 0 we get 1+that function to the power of 1 over that functon[wich is e] and that function again to the power, and if any other power remains(usually dose) you just to the reamining limit
(if someone is reading this and i just spoiled your joy of finding a solution to some problem,sorry)
do not ask me , maby he felt sadistic or thought we cannot do it
we found the easy way to solve for m in the following eq( a*sin(x)+b*cos(x)=m[probem was so find the min and max values for m given a and b(our pik, and the easy solution my colleague found is to just use tg(x/2)=t and the rest is math)]) so we are not that uncooperative, but if we have no idea were to look, anyway, thanq for at least trying to give me some direction on the solution will keep update if found solution solo or whatnot,

later edit, hold the phone, that pdf looks promising, leme just read it, may take a while

Last edited by rete (2013-07-13 05:39:57)

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#12 2013-07-13 05:42:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

Getting the limit of a function is not the same as getting the limit of a series like that. You will have to do a definite summation first.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#13 2013-07-13 06:00:40

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

ok, read the pdf and you are right good sir, half of it is gibberish, getting real numbers and stuff, and not even pretty ones, how come?
anyways, here is hoping
bobbym, i really appreciate you spending some time trying at least to point me in the right direction,
for me is just like a bloodhound finding meat , 1 morsel at a time, and you just said:" here boy here, the big stake is inside the pen, "
I just go wof and find the way to get in

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#14 2013-07-13 06:09:10

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

getting real numbers and stuff, and not even pretty ones, how come?

Things never change, do they?

School has the bad side effect of teaching students that all problems have answers that look like 2, 3, 5, π, √2 and i.
Nothing could be further from the truth. Real life problems have ugly answers that look like the ones in the pdf. Few, very few problems have pretty answers.

For instance:

Looks easy doesn't it? Ask some teacher you hate for the answer. Bet you a dollar he will be in here asking me when he fails. The reason is that most integrals, sum, recurrences, differential equations do not have closed form solutions. But all of them have those ugly solutions.

I ain't saying that there isn't some weird trick that will do your problems, olympiad problem are like that. Kids, come away saying wow, math sure is powerful. Truth is the problems are created to be solved by standard math methods. Those methods work on only a fraction of the possible problems there are.

Those ugly, gibberish methods do about 1000 times as many problems as the ones mathematicians love and keep under their pillow.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#15 2013-07-13 06:44:39

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

you are absolutely right,
i got nothing but a bunch of sin and x^4th power, and i do beleve what i am doing is wrong,
but still , i will try and solv this integral, who knows maby i will get lucky and do it , and i hope it is equal to something interesting big_smile
new stuff to learn
later edit:
ow god please do not tell me it is equal to pi/2

Last edited by rete (2013-07-13 06:50:49)

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#16 2013-07-13 06:55:45

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

I would suggest you let that integral go. That one has been constructed to beat all the known methods including the numerical ones. It requires many tricks to get the job done.

I will post if I find a standard solution to your sums.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#17 2013-07-15 19:21:31

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

ok, after some digging i have found a somewhat close problem, but i still do not understand how did they come up with the answer:
lim of n tends to infinity of sum from k =1 to n of 1/k(k+1)
this is the problem, and the solution given by the book is
you take out the sum and
sum of k=1 to infinity of 1/k -1/(k+1)
wich i get, you just split the terms since you have multiplication not summation
then that equals to 1-1/(n+1)
how do you get that summ?!?!
and the last step lim of a constant = constant and lim of 1/(n+1)=0 witch i get

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#18 2013-07-15 19:33:18

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

Hi;

The technique is called telescoping.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#19 2013-07-15 19:52:06

rete
Member
Registered: 2013-07-13
Posts: 12

Re: limits of a sum

bobbym(my hero)

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#20 2013-07-15 22:41:09

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

Hi;

Glad to help. Post if you need anything else.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#21 2013-07-16 00:23:10

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,657

Re: limits of a sum

bobbym wrote:

Hi;

The technique is called telescoping.

Of course, when telescoping you have to be careful. I remember a summation from gAr's thread that couldn't just be plainly telescoped.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#22 2013-07-16 00:31:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,562

Re: limits of a sum

Hi;

I remember that sum, Luckily, this one telescopes nicely.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

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