Convergent Series

Back on the 10th, we were given the assignment to find the value of 3 different sums (series).

The first of which was the sum of (1/(2^i)) as i goes from zero to infinity. The number that this series will eventually converge to is 2.

Next we had to find the sum of (1/(2^i)) as i goes from 1 to infinity. This converges to 1.

The last series we were assigned to evalute was the sum of (1/(2^i)) as i goes from 2 to infinity. As far as I can see this should converge to 1/2.

Proof:

1)

The n-th partial sum for this series is defined as
S _n = 1/2 + 1/2 ² + 1/2 ³ + ... + 1/2 n

If we divide the above expression by 2 and then subtract it from the original one we get:

S _n - 1/2 S _n = 1/2 - 1/2 ⁿ⁺¹

Hence, solving this for S _n we obtain

S _n = 2 (1/2 - 1/2 ⁿ⁺¹)

This is now a sequence, and we can take the limit as n goes to infinity. By our result on the power sequence, the term 1/2 ⁿ⁺¹ goes to zero,
so that lim S _n = 1

Taken From:
Interactive Real Analysis, ver. 1.9.4
(c) 1994-2005, Bert G. Wachsmuth

2) For the same series but with n starting at 0 or 2 we can just add or subtract the singleton element from our series limit. i.e. if we start with n=0, then one more element will be added to the series. The element 1/(2^0) = 1, so the limit as n goes to infinity is 1+1=2. On the other hand if we start with one less element such that n=2, then we can subtract the element 1/(2^1)=1/2, thus our limit would be 1-1/2=1/2.