2007-10-17 - More work with geometric/telescoping series and partial fractions

Today's nonsense
- Visitor! Mr. Warman? An awesome name.

Homework.

So far we have discussed: if it's telescoping, that's the one that looks like a fraction minus a fraction, and you write out the terms and they cancel, or partial fractions where we figure out the fractions and write out some terms and figure out the sum. The geometric formula could look misleading, because it's not the "a" in the formula but instead the first term (which may be defined differently, so check the summation symbol for that information). Let's try some more of "find the sum if it exists."

ex 1) sum from n=2 to infinity of 1/(n^2 -1) ... and so it's either a geometric series, telescoping series, or partial fractions, and so it's going to be partial fractions.

How do we apply this to repeating decimals? Say that we have 0.080808.... what then? You break it up into 8/10^2 and 8/10^4 and 8/10^6 etc.

Another box. Most of those homework problems will be geometric or telescoping because that's what we've covered so far.

Limit of n^th Term of a Convergent Series

If the series sum of a_n converges, then the sequence {a_n} converges to 0.
If the sum of all of the terms converges towards a specific value, then the sequence, the individual terms, converge to zero, so they are getting smaller and smaller. So the sequence is just what they end up at, but the series is what you do if you add them all up. If the series converges, that means the sequence, the terms themselves, must be getting closer to zero. If there is a sum that it heads towards, then the individual terms must be getting closer to zero.
-- If the sum converges to a specific value, then the terms get closer to zero.


First Test for Convergence

n^th Term Test for Divergence

If the sequence {a_n} does not converge to 0, then the series sum of a_n is going to have to diverge, if you're just adding + 1 + 1 + 1 etc., then what's that going to do? If it does not go to zero, then the series diverges. So take the limit of the sequence. You always start with the nth term test. If the limit is not zero, it diverges. If the limit is zero, you need to try another test.

A ball is dropped from 6 ft. The height of each bounce is three-fourths the present height. Find the total vertical distance traveled by the wall.