This week we had our test 2 and started looking into sequences. Next week, we will continue with properties of sequences and start with series.
Test 2 focused on general techniques of integration, application of integrations and improper integrals. From the questions I got from you guys, the hardest step is to start a problem and choose which method to use. I suggest you do the problems in section 7.5, as those problems use all the techniques we ever discussed this semester and you need to make a judgement.
For sequences, we care about convergence and divergence as n approaches to infinity. There are some general ideas to help you determine the convergence of a sequence:
- Look at corresponding continuous functions. As long as the function has a horizontal asymptote as x approaches to infinity, we can conclude the convergence of sequence. Note the divergence of function *does not* conclude the divergence of the sequence! The reason of looking at continuous functions is that we can use properties of functions that discrete sequences don’t have, such as L’Hospital rule.
- Use the properties of limit laws. If we have two convergent sequences, we can conclude the limits of their sums, differences, products, quotients, etc. We can use these properties to chop problems into small pieces and deal with each piece individually.
- Compare with known sequences (Squeeze Theorem). Same for all comparison rules, try some values and get a general direction before you charge ahead to find bounding sequences.
- In general, if you don’t know how to deal with infinities, try to convert them into zeros.
Next week, we will start looking at “sums” of the sequences, namely, series. You will use series in all sorts of applications, and they have interesting properties themselves as well!
Reading: 11.1, 11.2