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January 29th, 2008, 07:15 PM   #1
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Geometric series/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 (4/7)

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?
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January 30th, 2008, 12:50 AM   #2
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Quote:
Originally Posted by clooneyisagenius
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.
Let S =1 - (3/4) + (9/16) - (27/64) + ...,
so -(3/4)S = - (3/4) + (9/16) - (27/64) + ... = S - 1,
and so S = 1/(1 + 3/4) = 4/7.

You can find n such that (3/4)^n < 0.001. For example, n = 25. Can you see why such n has the property requested?
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