Decide whether each of the following statements is true or false. Justify your answer in each case by giving an example or stating any general result seen at the lecture: Mathematics, Coursework, NU, UK

1 . Consider the sequence (Xn) defined by

Using the definition of convergence, prove that (xn) does not converge to 1/4.

2. Using the Combination theorem and/or the Sandwich rule show that each of the following sequences converges and find their limits.

3. Decide whether each of the following statements is true or false. Justify your answer in each case by giving an example or stating any general result seen at the lecture, as appropriate.

(a) There exists a sequence (n) satisfying an > 8 for all n and converging to 8.
(b) If a sequence is divergent then it must be unbounded.

4. Consider the sequence (an) defined by a1 = 1 and

5.  Using partial sums, show that the series

is convergent and find its sum.

Answer

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