Jan 20| Feb 1 | Mar 1 | Apr 5 | Phy 515 page
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20 Jan
Definition of the convergence of an infinite series. Example: the
geometric series both finite and infinite. Use this together with the
comparison test to find the ratio test for convergence.
The harmonic series as an example of something that doesn't converge,
demonstrated by explicitly getting a lower bound for the size of some
partial sums.
The integral test. Derive it graphically and indicate how it can be
used to get numerical approximations to series.
22 Jan
The Riemann zeta function definition. Graph it for x > 1. The Gamma
function definition. Use parametric differentiation to show that is a
factorial for positive integer values of the argument.
How to improve the convergence of a series: Use the example of
z ( 2 ) and combine it with the series sum of
1/ n( n+1 ), which has the known sum of 1. This trick makes the new
series converge as 1/ n3 instead of 1/ n2.
Alternating series. Show the simple test for convergence: the terms are
monotone decreasing and alternating is all that it takes. You can use
this idea to estimate the error if you stop the series after a finite
number of terms. Define absolute convergence and show by an example
that a non-absolutely converging series can be re-arranged to converge to
anything you want.
Show how to get the series for a logarithm by integrating 1/ ( 1 + t ).