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Today:
Your "redo" should be very neat and tidy. You may work with with others, but your final work, on your own test, should be your own.
These are due Thursday of next week. Hand them in attached with your original exam.
But we do need to take care of some serious errors....
we define a new sequence :
These are called partial sums. If
Then we say that
exists, and is equal to L. More formally,
Now geometric series are sufficiently important that it's useful to include that special case:
There's a really fun proof of the first part above, based on a trick. Let's have a look at it.
This theorem tells us that if the terms of a sequence aren't asymptotic to the x-axis, we can forget about the partial sums converging.
Finally, series behave the way we'd hope (provided they're convergent):
Theorem 3 is just a corollary of Theorem 2, where the integrals are the obvious power functions:
This theorem says that, in the long run, one series has terms which are simply a constant times the terms in the other series.