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This is a guide to help you prepare your own summary of convergence methods for series. The best way to use this guide is in study groups trying to explain to each other.
(1) Answer the following questions, they will help you clarify the concepts you will work with: (a) What is the meaning of the symbol
k=1 ak, also called an “infinite series”?
(b) What is an n-th partial sum for the infinite series
k=1 ak?
(c) How is the sequence of partial sums {Sn} different from {ak}?
(d) What is meant by “the sum” of an infinite series?
From now on, you may use table 8.1 (Review of convergence Methods) that appears in your textbook on pages 540-541.
(2) Geometric series. (a) Write the general form of a geometric series: (b) A geometric series
- Converges if:
- Diverges if:
- If a geometric series converges, its sum is: (c) Invent an example of a geometric series, identify a and r in your example and decide whether your series converges or diverges.
(3) Telescoping series: (a) Write the n-th partial sum of a telescoping series and show how its telescoping.
(b) What examples of telescoping series were studied in class? Try to do those examples on your own and try to under- stand why the series are telescoping.
(b) In order for one to be able to use the integral test for the series
k=1 ak^ one must verify: (i) ak ≥ (ii) ak is (c) Usually, one verifies that the sequence ak is decreasing by showing either of the following: (i) ak+1 ak directly (ii) The associated function f (x) satisfies that f ′(x) 0 (iii) If one is familiar with the graph of the associated function f (x), one can just draw the graph and it will tell that the associated function, and therefore the sequence, is decreasing. (d) What improper integral do you need to analyze in the in- tegral test?
(e) What does the integral test say?
(f) Do some examples. Note: If the improper integral that you get by replacing k by x is easy to work with (say, there is an obvious substitution or such) then the integral test may work, but you have to verify that you can use it ( 5b).
(6) Divergence test. (a) The divergence test says that if the infinite series
k=1 ak converges then lim k→∞ ak = (b) If lim k→∞ ak = 0, will the series
k=1 ak^ necessarily converge? Give an example where this is not true.
(c) If lim k→∞ ak = 0 and the series
k=1 ak^ does converge will the series converge to zero as well? Give an example that shows that this is not true.
(7) Comparison Test (a) The comparison test says that (i) If 0 ≤ ak ≤ bk and
bk then
ak (ii) if 0 ≤ dk ≤ ak and
dk then
ak (b) Give examples where the comparison test is useful to decide the convergence or divergence of a series.
