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clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Intro to Analysis - Find target of series, etc.

I need to:

Find the target of the series: 
1 - (3/4) + (9/16) - (27/64) + ... + (-1)^k*(3^k/4^k)
Then i need to find any value of n so that any partial sum with at least n terms is within 0.001 of the target value. Justify the answer.

So here's what I did. I made a program on Mathematica that gave me that the target value of the partial sum is .571429

I also found that at the sum from 0 to the 22nd term is equivalent to 0.572193

I suppose I have my answer but I cannot justify it - except using proof by mathematica program.  I am worried though that when I have an exam and am forced do this without mathematica I will not be able to.  Is there any advice? Or can someone teach me how to do this?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Intro to Analysis - Find target of series, etc.

The key, as you hinted at, is in not using mathematica.  First, prove that:

Now apply this to the entire summation, and look up (or recall) the term "geometric series".

Don't do what I did.  Make sure you replace this summation with a 2k in the exponent.  This is because you are taking up two terms.

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"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..."

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Re: Intro to Analysis - Find target of series, etc.

AHA!

So, I know how to find that the target value is 4/7 (or .572193) now... (I THINK!)

I want to look at the geometric series:

and plug in

for x
which gives me:

Which equals 4/7! Thanks Ricky!

Now... to find the value of

so that any partial sum with at least

terms is within 0.001 of the target value.

Last edited by clooneyisagenius (2008-01-29 16:44:03)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Intro to Analysis - Find target of series, etc.

5000000 should do the trick.  But once you put it as a positive sequence (like I did), estimating the error value should be rather simple.

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"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..."