Advanced Calculus: Problem sheet 3

Solutions should be submitted to tutors by 1pm on Monday 5 March 2001

1. Find the general solutions of the following differential equations.

(a) x^{2}y′′ + 4xy′ + 2y = 0;

(b) x^{2}y′′ + xy′ − 4y = 3x;

(c) x^{2}y′′ − 3xy′ + 4y = x ln x.

2. Give yes/no answers to the following questions. Justify your answers as far as possible:

(a) Is every bounded sequence convergent?

(b) Is every convergent sequence bounded?

(c) Is every nondecreasing sequence bounded below?

(d) Is every nondecreasing sequence bounded above?

3. I gave a definition of what it means for a sequence {a_{n}} to diverge to ∞. Give a definition of what it means for a sequence {a_{n}} to diverge to −∞.

The following problems are adapted from Chapter 8.1 of the course textbook (Thomas and Finney, Calculus, 9th edition). See the book for more information and examples.

4. In each case find the values of a_{1}, a_{2}, a_{3} and a_{4}.

(a) a_{n} = 2n/(3n − 1);

(b) a_{n} = 1 + (−1)^{n};

(c) a_{n} = (2^{n} + 1)/2^{n}.

5. Define a sequence {a_{n}} recursively by letting a_{1} = 0 and a_{n+1} = a_{n} + 1/(n(n + 1)). Write out the first ten terms of this sequence.

6. Define sequences \{a_{n}\} recursively by letting a_{1} = 2 and a_{n+1} = \frac{1}{2}(a_{n} + 3/a_{n}). Using your calculator, compute the first ten terms of this sequence. To what number does this sequence appear to converge.

Repeat for the sequence \{a_{n}\} defined by a_{1} = 1 and a_{n+1} = \frac{2}{3}(a_{n} + 1/a_{n}^{2}).

7. Find a general formula for the nth term of the following sequences:

(a) 1, −\frac{1}{8}, \frac{1}{27}, −\frac{1}{64}, \frac{1}{125}, …, [reciprocals of cubes of positive integers with alternating signs];

(b) 3, 7, 11, 15, 19, … [every other odd positive integer];

(c) 2, 6, 12, 20, 30 … [products of pairs of consecutive numbers].

8. Determine which of the following sequences {a_{n}} are nondecreasing and which are nonincreasing. Also determine which are bounded. [Hint: it may help to consider the differences a_{n+1} − a_{n} or the ratios a_{n+1}/a_{n} of consecutive terms.]

(a) a_{n} = (4n + 1)/(n + 1);

(b) a_{n} = 2^{n}/n!;

(c) a_{n} = 1 − 2/n + 1/n^{2}.

9. Which of the following sequences {a_{n}} converge and which diverge? Find the limit of each convergent sequence.

(a) a_{n} = \frac{n^{2} − 1}{n^{2} + 1};

(b) a_{n} = \frac{n + (−1)^{n}}{n + 1} ;

(c) a_{n} = \frac{1 − 2n^{2} + 3n^{4}}{n^{4} + 6n^{3} + 1} ;

(d) a_{n} = \frac{(−1)^{n}n}{n + 1};

(e) a_{n} = \frac{cos n}{\sqrt{n}};

(f) a_{n} = \frac{ln n}{n + 1};

(g) a_{n} = \frac{ln 2n}{ln n};

(h) a_{n} = \frac{n^{n}}{n!} ;

(i) a_{n} = (1 − \frac{3}{n})^{n};

(j) a_{n} = (\frac{3^{n}}{n^{2}} )^{1/n}.

RJC 19/2/2001