This exercise sheet contains some additional questions that may be useful for revision or review of the material in these web pages.
This is work in progress. Please return here later.
Exercise.
Using the theorem on subsequences, and/or the theorem on boundedness
of convergent sequences, and/or the theorem bounding the limit of a bounded
sequence, prove the following sequences (_{
)
} with
(b) _{ =3+(-1) }.
(c) _{ =( + 4 ) }.
Exercise.
Prove that
Hence show that the series
Exercise.
A function
Prove that
(Hint: Split into cases and divide top and bottom of the
fraction by appropriate numbers. You may find it helpful
to use that fact that
Solution to Exercise 2, in the special case when 0
The next two exercises concern some common misconceptions and
errors in beginning analysis. Many of these examples are worth learning.
Some are easy but some are rather subtle and tricky. (One or two are
dreadful howlers that I hope no-one reading this would ever consider
writing.) If you have a good feel
for things like this you are
well on the way to being rather good at analysis.
Exercise.
The following statements are all false. In each case, find a counterexample.
(a) If the sequence (_{ ) } diverges and the sequence (_{ ) } diverges then their sum (_{ + ) } diverges.
(b) If the sequence (_{ ) } diverges and the sequence (_{ ) } diverges then their product (_{ ) } diverges.
(j) If
(k) If
Solution to Exercise 3(a): Let the sequence (_{ ) } be defined by _{ =(-1) } and (_{ ) } be defined by _{ =(-1) +1 }. Then both these sequences diverge, and yet their sum is _{ + =-1+1 } or 1+-1 which defines the constant sequence with value 0, which converges.
Solution to 1(a): Let (_{
)
}
be the subsequence of (_{
)
} consisting of all those
_{
} with
Therefore (_{
)
} does not converge, for if _{
}
as