Infinity and Beyond 4: Sums

One of the first ways we experience infinity is through sums.

1+1+1+1+1 ... = ∞

Any sum that goes out to infinity is said to DIVERGE.  Let's look at some examples

HARMONIC SERIES:

1 + 1/2 + 1/3 + 1/4 + ... = ∞

Proof: 

1 + (1/2) + (1/3+1/4) + (1/5+1/6+1/7+1/8) + (1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16) + ...

Each sub-sum is greater than or equal to 1/2, so the sum is greater than

1 + 1/2 + 1/2 + 1/2 + ... = ∞

Power of two series:

1 + 1/2 + 1/4 + 1/8 + ... = 2

This can be seen by cutting a line segment over and over again (One piece will have size 1, the next 1/2, etc)


A very famous theorem says that if we define:

Z(s) = 1^-s + 2^-s + 3^-s + 4^-s + ...

Z(s) diverges when s < 2.

(The next post will give more examples of converging series)

Here is a fun problem using the above setup:

The tail of a giant wallaby is attached by a giant rubber band to a stake in the ground. A flea is sitting on top of the stake eyeing the wallaby (hungrily). The wallaby sees the flea leaps into the air and lands one mile from the stake (with its tail still attached to the stake by the rubber band). The flea does not give up the chase but leaps into the air and lands on the stretched rubber band one inch from the stake. The giant wallaby, seeing this, again leaps into the air and lands another mile from the stake (i.e., a total of two miles from the stake). The flea is undaunted and leaps into the air again, landing on the rubber band one inch further along. Once again the giant wallaby jumps another mile. The flea again leaps bravely into the air and lands another inch along the rubber band. If this continues indefinitely, will the flea ever catch the wallaby? (Assume the earth is flat and continues indefinitely in all directions.)

Answer:

For this question, I defined a unit to be the ratio between a mile and an inch.  When the flea first jumps one inch onto the rubber band, for every mile the wallaby jumps, the flea is pulled forward one inch.  We can therefore say that the flea has moved forward one unit.  Now, when the flea jumps another inch, for every two miles the wallaby jumps, that jump will expand one inch.  We can say the flea jumped a ½ unit.  Using this concept, we can say, when the flea jumps 

1 + 1/2 + 1/3 + 1/4 + ... 

units, does the flea reach 63360 units?  Because the harmonic series diverges, the answer is YES