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LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

How to find the limit of a sum of a series?

Im doing my homework on Infinite Series and I need help how to find the limits of the sums of the series.

For example:

So I know that a = 1/4 and r = 1/4 and since:

The series converges.

This was just an example. How do I find the limits of such series?

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: How to find the limit of a sum of a series?

I only know how to do this in the case of a geometric series.

The sum of a geometric series is given by:

If

, then as n goes to infinity,

becomes negligible.

Therefore, the limit of the sum of an infinite geometric series is given by

Last edited by Daniel123 (2008-03-21 10:50:28)

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: How to find the limit of a sum of a series?

Thanks a lot Daniel123.

The thing is that we havent got to those formulas in the book. They want us to find its limit by actually taking the limit of the series.

This how they do it in the solutions manual:

which gives the same answer as you specified but I'd like to do it this way but to be honest, I have no idea.

Last edited by LuisRodg (2008-03-21 11:01:21)

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: How to find the limit of a sum of a series?

Right OK I see what they've done. It seems a bit silly, as this is the exact same process used when deriving the general formula for the sum - if you have to do this for a lot of questions it will get extremely tedious. Maybe they're trying to get you to spot the formula for yourself?

Let the sum of the series inside the square brackets be denoted by

, which is the formula I stated earlier.

Taking the 1/4 out originally seems pointless. The only reason I can think of is so that you end up with an expression for the sum that looks like the actual formula.
If you had left the 1/4 in then you would end up with the same answer in the end, but a slightly different expression for the sum above (with n+1 instead of n and 1/4 in place of the 1 in the numerator).

From here,

Last edited by Daniel123 (2008-03-22 13:29:50)

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: How to find the limit of a sum of a series?

I was able to follow you but I feel that if I try to do it for any other that I wouldnt know how to continue. Could you please explain where did you get the idea of doing all those steps? Like I said, I understand them but I wouldnt know how to develop this steps for another problem.

I just decided to use the formula to solve them but I ran into some which I couldnt find a ratio....So I guess the series is not geometric?

For example:

and also this one:

Last edited by LuisRodg (2008-03-23 03:24:35)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How to find the limit of a sum of a series?

Those two sums need to be solved by a different method.

Method of telescoping sums

In the case of

put

into partial fractions first.

Last edited by JaneFairfax (2008-03-23 04:18:54)

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: How to find the limit of a sum of a series?

hint: what is the sum of two following terms, ie term n and term n+1?

Last edited by Kurre (2008-03-23 04:33:50)

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: How to find the limit of a sum of a series?

I was able to follow you but I feel that if I try to do it for any other that I wouldnt know how to continue. Could you please explain where did you get the idea of doing all those steps? Like I said, I understand them but I wouldnt know how to develop this steps for another problem.

As I said, this is exactly the same method for deriving the formula. I know the method off by heart from having to prove it, so I just replaced a and r with the numbers I needed.

From here it would be useful to take rSn away from Sn, as most of the terms will cancel out, making it easier to find the sum.

Those are the steps I had in my mind when finding the sum, but I had never used it with numbers before. I have no idea if this is how you are supposed to do it, but I can't see any other way? Pehraps someone else has an idea?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: How to find the limit of a sum of a series?

Almost in every case in mathematics, a solution is first suggested before it is derived/proven.  This is how you should approach this problem as well.

s1 = 1/4
s2 = 5/16
s3 = 85/256

It should be pretty obvious what s4 is going to be from just computing s1, s2, and s3.  s4 = (85*4+1) / 512.  Now we need to prove this.  And we do so by induction, and it should be fairly straightforward.  Once that's done, we have a sequence for the partial sums and we evaluate it's limit to see where this sequence converges to.

Moral: It's a lot easier to get where you're going once you know where you want to go.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: How to find the limit of a sum of a series?

How do I use this?

Can you give me an example of how to apply this?

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: How to find the limit of a sum of a series?

the function is then

because you then have:

giving:

which is obviously

Last edited by luca-deltodesco (2008-03-24 05:46:42)

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The Beginning Of All Things To End.
The End Of All Things To Come.

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: How to find the limit of a sum of a series?

Thanks a lot.