Sheet 5 Moodle version

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Q1

For \( n \in \mathbb{N} \) let \( \displaystyle a_n = \int_1^n \frac{\cos x}{x^2} \mathrm{d}x \).  Prove that for \( m \geq n \geq 1 \) we have \( |a_m - a_n| \leq n^{-1} \) and deduce that \( (a_n) \) converges.

By integration by parts, or otherwise, demonstrate the existence of \( \displaystyle \lim_{n\to \infty} \int_1^n \frac{\sin x}{x} \mathrm{d}x \).

Q2

(a) Let \( p \) be a positive integer.  By considering the partial sums, prove that \( \displaystyle \sum_{k\geq 1} \frac{1}{k(k+p)} \) converges.  What is \( \displaystyle \sum_{k=1}^{\infty} \frac{1}{k(k+p)} \)?

(b) Evaluate the sum \( \displaystyle \sum_{k=1}^{\infty} \frac{\cos k}{2^k} \).

(c) Use the Comparison Test to prove that \( \displaystyle \sum_{k \geq 1} \frac{2k+1}{(k+1)(k+2)^2} \) converges.

Q3

Let \( (a_n) \) be a sequence of real or complex numbers, and assume that \( \displaystyle \sum_{k=1}^{\infty} |a_k| \) converges.  Let \( s_n = a_1 + \dotsb + a_n \) and \( S_n = |a_1| + \dotsb + |a_n| \).  By considering the partial sums \( s_n \) and \( S_n \), prove that

\[ \left| \sum_{k=1}^{\infty} a_k \right| \leq \sum_{k=1}^{\infty} |a_k|. \]

[You may assume the triangle inequality, and absolute convergence implies convergence.]

Q4

The number known as \( \mathrm{e} \) is defined by

\[ \mathrm{e} = \lim_{n \to \infty} s_n, \quad \textrm{ where } s_n = \sum_{k=0}^n \frac{1}{k!}. \]

(a) Prove that \( (s_n) \) is increasing and bounded above.  Deduce that the limit defining \( \mathrm{e} \) exists.  [Hint for getting an upper bound: compare \( s_n \) with the sum of a geometric progression.]

(b) Show that, for \( n \geq 1 \),

\[ 0 < \mathrm{e} - \sum_{k=0}^n \frac{1}{k!} < \frac{1}{n!n}. \]

Deduce that \( \mathrm{e} \) is irrational.

Q5

There is a real number \( L \) such that  \( \displaystyle \left(1 + \frac{1}{n}\right)^n \to L \) as  \( n \to \infty \) (see the example in Section 25 of the lecture notes).  In fact \( L = \mathrm{e} \) (see the Supplementary notes on e by HA Priestley).  Assuming this fact, show that

\[ \left(1 - \frac{1}{n}\right)^n \to \frac{1}{\mathrm{e}} \quad \textrm{ as } n \to \infty. \]

Q6

(a) Prove that \( \displaystyle \sum_{k \geq 1} (-1)^{k-1}(\sqrt{k+1}-\sqrt{k}) \) converges.

(b) For \( n \geq 1 \), let \[ s_n = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dotsb + \frac{(-1)^{n+1}}{2n-1}. \]

Show that we can use the Alternating Series Test to prove that \( (s_n) \) converges to some limit  \( L \).  By examining the proof of the AST, prove that \( \frac{2}{3} < L < \frac{13}{15} \).

Q7

Let \( \sum a_k \) be a series of real numbers.  Which of the following are true and which are false?  Give a proof or counterexample as appropriate.

1. \( k^2 a_k \to 0 \) implies \( \sum a_k \) converges.
2. If \( \sum a_k \) converges, then \( \sum (a_k)^2 \) converges.
3. If \( \sum a_k \) converges absolutely, then \( \sum (a_k)^2 \) converges.
4. \( \sum (a_k)^2 \) convergent implies \( \sum (a_k)^3 \) convergent.

Q8

(Optional, and more challenging) For each of the following statements, give a proof or a counterexample.

(a) For a divergent series \( \sum a_k \) of positive terms, \( \displaystyle \sum \frac{a_k}{1+a_k} \) is also divergent.

(b) Assume that \( a_k > 0 \), and write \( s_k = a_1 + \dotsb + a_k \).  Then \( \sum a_k \) and \( \displaystyle \sum \frac{a_k}{s_k} \) either both converge or both diverge.