Two Infinite Series That Add Up to 1

Oct 2013
719
91
New York, USA
I watched a video on the sum of the infinite series 1/(1*2) + 1/(2*3) + 1/(3*4) + ... and each term can be broken down into two parts. The first term = 1 - 1/2, the second term is 1/2 - 1/3, and it adds and subtracts the same fractions so the sum is 1 - 1/(k(k+1)). As k approaches positive infinity, that fraction approaches 0, and the sum is 1. Another infinite series is that 1/x + 1/(x^2) + 1/(x^3) ... = 1/(x-1). When x = 2, this becomes 1/2 + 1/4 + 1/8 + ... = 1. If two series = 1, then they equal each other. Setting them equal produces this:

1/2 + 1/6 + 1/12 + 1/20 + 1/30 + ... = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +...

I showed enough terms to get to the first time when the term on the left is greater, which is the fifth term on each side. Adding the first four terms on the left is 4/5, and adding the first four terms on the right is 15/16. 15/16 - 4/5 = 11/80. Therefore if you start each sequence with the fifth term, it looks like this:

1/30 + 1/42 + 1/56 + ... = 11/80 from the first four terms being greater + 1/32 + 1/64 + 1/128 ...

Can you work with infinite series like that?
 

skipjack

Forum Staff
Dec 2006
21,493
2,476
Yes, you just have, but take care with a series such as 1 - 1 + 1 - 1 + . . . (for obvious reasons).
 
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