Expert: Scotto - 8/5/2010

Question
Consider an infinite series of the form

(+-)3(+-)1(+-)1/3(+-)1/9(+-)1/27(+-)....(+-)1/3^n(+-)...

The number 3,1, etc. are given but you will decide what the signs should be.
a)Can you choose the signs to make the series diverge?
B)Can you choose the signs to make the series sum to 3.5?
c)Can you choose the signs to make the series sum to 2.25?
d)Can you choose the signs to make the series sum to 4.4?
In each case, if your answer is Yes, then specify how to choose the signs; if your answer is No, then explain

Answer
Note that 3 to powers is 3, 9 (3^2), 27 (3^3), 81 (3^4), 243 (3^5), etc.  This will be used later.

a) Always convergent.  It is known that sum(1/3)^n starting with n = 0, so leaving off the leading 3, gives a sum of 1/(1 - 1/3) = 1/(2/3) = 3/2.  Now when we add in the leading 3, we get 3/2 + 6/2 = 9/2.  So if some minus signs are thrown in, that is the upper bound.

b) Yes.  Take the 1st two term as +1 and from the 2nd on, make them all -1.

Since we know the series starting with n=0 adds to 3/2, then if we multiply each term by 1/3, the sum would be 1/2.  If we start with
-1/3 as the first term, this does it.
Yes, the series is 3 + 1 - 1/3 - 1/9 - 1/27 - 1/81 - 1/243 ...,
for this come out to be
4 - (1/3)(1 + 1/3 + 1/9 + 1/27 + ...), and that is
4 - (1/3)(1/(1 - 1/3)) = 4 - (1/3)(3/2)= 4 - 1/2.

c) Yes.  If the terms are taken alternatively as plus or minus 1,
we get 3 - 1 + 1/3 - 1/9 + 1/27 - 1/81 ...
This cae be rewritten as 2 + 2/9 + 2/81 + ...
This is the same as 2(sum(1/9)^n) = 2(1/(1 - 1/9)) = 2(1/(8/9)) =
2(9/8) = 9/4 = 2.25.

d) No.  If all of the terms are positive, the result is 4.5.
If we take the 1/9 as positive, the summation is too much.
If we take the 1/9 as negative, the summation is always too little.
So even though 4.4 is in the range, it can't be gotten.

See, if we take the first four terms as +1, we get 3 + 1 + 1/3 + 1/9,
and that is 4.44444....., which is bigger than 4.4.
If we take the first four terms as 3 + 1 + 1/3 - 1/9, we get
4.22222..., and that is .17777777 to low, which can't be recovered.

This is because we start with the next term being 1/27, so factoring out 1/27 from all the terms gives (1/27)sum(1/3^n), which is known to be (1/27)(1/(1 - 1/3)) = (1/27)(1/(2/3)) =  (1/27)(3/2) = 1/18,
and 1/18 is 0.05555555...