Sheet 7 Moodle version - Updated in 2023

This is the same content as the pdf in the Sheet 7 assignment, but visible as a Moodle page.

Q1

(a) Prove that

\[ 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \dotsb = \frac{3}{2}\log 2. \]

(b) Calculate the value of

\[ \sum_{k=1}^{\infty} \frac{1}{k(9k^2-1)}. \]

Q2

Find the radius of convergence of each of the following real power series (start the sum at \( k=1 \) where appropriate):

1. \( \displaystyle \sum(-1)^k k^2 x^k \);
2. \( \displaystyle \sum \frac{(2k-1)(2k-3)\dotsm 3 \cdot 1}{k!}x^k \);
3. \( \displaystyle \sum \left(\frac{x}{k}\right)^k \);
4. \( \displaystyle \sum k^{\frac{1}{k}} x^k \).
   

Q3

(a) Give examples of real power series \( \sum c_k x^k \) with the specified radius of convergence \( R \) and with the specified behaviour at  \( \pm R \):

1. \( R = 1 \) and the power series converges at \( -1 \) but diverges at 1;
2. \( R = 1 \) and the power series converges at 1 and diverges at \( -1 \);
3. \( R = 2 \) and the power series converges at 2 and \( -2 \);
4. \( R = 2 \) and the power series diverges at 2 and \( -2 \).

(b) Let the real series \( \sum a_k x^k \), \( \sum b_k x^k \) and \( \sum c_k x^k \) have radius of convergence \( R \), \( S \) and \( T \), respectively, where \( c_k = a_k + b_k \).  Obtain a lower bound for \( T \) involving \( R \) and \( S \).  Provide examples to illustrate what possibilities can arise.

Q4

For which real values of \( x \) does \( \sum x^k \) converge?  Use the Differentiation Theorem for power series to evaluate

1. \( \sum_{k=1}^{\infty} kx^k \);
2. \( \sum_{k=1}^{\infty} k^2 x^k \),

specifying where the formulae you obtain are valid.

Q5

(a) Prove that the power series

\[ \sum_{k=0}^{\infty} \frac{x^k}{(2k)!} \]

and

\[ \quad \sum_{k=0}^{\infty} \frac{x^k}{(2k+1)!} \]

have infinite radius of convergence.

(b) Define

\[ p(x) = \sum_{k=0}^{\infty} \frac{x^k}{(2k)!} \]

and

\[ \quad q(x) = \sum_{k=0}^{\infty} \frac{x^k}{(2k+1)!}. \]

Use the Differentiation Theorem to compute \( p'(x) \) and \( q'(x) \) and prove that \( 2p'(x) = q(x) \) and \( p(x) - q(x) = 2xq'(x) \).  Hence prove that, for all \( x \),

\[ (p(x))^2 = 1 + x(q(x))^2. \]

Q6  [New Question from 2023 -- Feedback is welcome: Alex Ritter, ritter@maths]

Show that:

1. Given any point \( p \) in an open subset \( U\subseteq \mathbb{R} \), there are rationals \( c_p,r_p\in \mathbb{Q} \) such that the interval \( (c_p-r_p,c_p+r_p) \subseteq U \) contains \( p \).

2. Any open subset \( U\subseteq \mathbb{R} \) is a countable union of open intervals.

Optional: One can ensure the intervals are disjoint (allowing unbounded intervals).

3. If \( S\subseteq \bigcup_{i\in I}U_i \) for open sets \( U_i\subseteq \mathbb{R}\), there is a countable \( J\subseteq I \) with \( S\subseteq \bigcup_{j\in J}U_j \).

4. If \( S \) in (3) is bounded and closed, then there is a finite such \( J \).

5. \( (0,1) \) is the union of countably many closed intervals, but it is not the union of countably many disjoint closed intervals.

Challenge Exercise: \( (0,1) \) is not the union of countably many disjoint closed sets.

Find the radius of convergence of the power series defining the function \( J_0 \), where

\[ J_0(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(k!)^2} \left(\frac{x}{2}\right)^{2k}. \]

Use the Differentiation Theorem to show that \( y = J_0(x) \) is a solution of the equation \(xy'' + y' + xy = 0 \).

Q8

(Optional) The Fibonacci numbers \( F_n \) are defined by \( F_0 = 0 \), \( F_1 = 1 \) and \( F_{k+2} = F_{k+1} + F_k \) for \( k \geq 0 \).  Define

\[ F(x) = \sum_{k=0}^{\infty} F_k x^k. \]

Determine the radius of convergence of the series defining \( F(x) \).

By summing the identity

\[ F_{k+2}x^k = F_{k+1}x^k + F_k x^k \]

over all \( k \geq 0 \), or otherwise, find \( F(x) \) in closed form.

Q9

(Optional: addition formula for \( \sinh \)) Show that each of the power series \( \displaystyle \sum_{k=0}^{\infty} \frac{x^{2k}}{(2k)!} \) and \( \displaystyle \sum_{k=0}^{\infty} \frac{x^{2k+1}}{(2k+1)!} \) converges for all \( x \in \mathbb{R} \).  Define

\[ C(x) = \sum_{k=0}^{\infty} \frac{x^{2k}}{(2k)!} \]

and

\[ S(x) = \sum_{k=0}^{\infty} \frac{x^{2k+1}}{(2k+1)!}. \]

(a) Assuming the Differentiation Theorem for power series, calculate the derivatives of \( C(x) \) and \( S(x) \).

(b) For fixed \( d \in \mathbb{R} \), let

\[ f_d(x) = S(d+x)C(d-x) + S(d-x)C(d+x). \]

By considering the derivative of \( f_d(x) \), prove that, for all \( a, b \in \mathbb{R} \),

\[ S(a+b) = S(a)C(b) + S(b)C(a). \]