No not Eastenders (as a tutor joked to me), but the topic of the OU's M208 book AA3. I have got through this in a fairly short space of time and I have already started the last book in this first block on analysis, continuity.
I really enjoyed AA3. I like numbers and I like sequences and series, some of which are a bit quirky at times. The good thing about this third book of analysis is that by the end of it I felt like everything was falling into place. The main emphasis is on being able to ascertain whether a series such as 1+1/2+1/3+1/4+... is convergent or not and by the end of the book all these techniques are at your fingertips. You find yourself using all the methods of AA2 on sequences as well as what is in this book, so you do feel like you are getting somewhere. I even felt that I could prove the convergence of the series for ex for x<0.
The only slight gripe I would have is that I would have liked to have seen some more work on actually determining the sum of a convergent series as I think this can be quite interesting. For example, we are told as an aside that the sum of the series 1+1/4+1/9+1/16... is π2/6, but it would be nice to know how this is obtained.
Friday, 25 November 2011
Friday, 11 November 2011
Bounded, unbounded, convergent, divergent.
I am getting on ok with M208's book AA2 on sequences and I haven't come across any problems apart from one or two minor niggly points about some solutions to the exercises that I will eventually have to ask about. One of the nice things about this book is that I think I have finally got my head around the differences between the meanings of convergent/divergent, bounded/unbounded and tends to plus or minus infinity.
A sequence that tends to a limit L where L is a real number is convergent. If L is zero then this is a null sequence. If {an} is a sequence and |an|≤K for all n, for some positive number K, then the sequence is bounded. If the sequence is not bounded, then it is unbounded. If the sequence is convergent then it is bounded. If the sequence is unbounded then it is divergent. A sequence that tends to plus infinity is both unbounded and divergent. The same applies to a sequence that tends to minus infinity.
A point that should be made is that the converse of the statement "If the sequence is unbounded then it is divergent" is not true. The converse is "If the sequence is divergent then it is unbounded" and this is not true because bounded sequences such as {(-1)n} are divergent but not unbounded. Note also that some sequences such as {(-1)nn} are unbounded but do not tend to either plus infinity or minus infinity.
I hope that makes it all clear for everyone. I can bet that in a month or two it will all become opaque for me again!
A sequence that tends to a limit L where L is a real number is convergent. If L is zero then this is a null sequence. If {an} is a sequence and |an|≤K for all n, for some positive number K, then the sequence is bounded. If the sequence is not bounded, then it is unbounded. If the sequence is convergent then it is bounded. If the sequence is unbounded then it is divergent. A sequence that tends to plus infinity is both unbounded and divergent. The same applies to a sequence that tends to minus infinity.
A point that should be made is that the converse of the statement "If the sequence is unbounded then it is divergent" is not true. The converse is "If the sequence is divergent then it is unbounded" and this is not true because bounded sequences such as {(-1)n} are divergent but not unbounded. Note also that some sequences such as {(-1)nn} are unbounded but do not tend to either plus infinity or minus infinity.
I hope that makes it all clear for everyone. I can bet that in a month or two it will all become opaque for me again!
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