Math Is Fun Forum

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Construct a series that suggests that the sum?

I need to construct a series that suggests that the sum (of alternating 1’s and −1’s) is 1/4. Or 2/3.

Here's what was written in class before this:
Here is an example that seems to suggest that
1 − 1 + 1 − 1 + 1 − 1 ± · · · = 1/3
.
Consider the series
S = 1 − x + x3 − x4 + x6 − x7 ± · · · .
Note that
S = (1 − x)(1 + x3 + x6 + · · ·) = (1 − x)*(1/(1 − x^3)) = (1/(1 + x + x^2))
Evaluate at x = 1.

.......

I'm not even sure how the example works completely... let alone how to do one on my own.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Construct a series that suggests that the sum?

You must consider the radius of convergence R as well. For your series S, R = 1, so S only converges for x with |x| < 1. When x = 1, S is clearly divergent, so, no can do.