Sheet 6 Moodle version

For the following choices of \( a_k \), use the indicated test to establish whether or not \( \sum a_k \) converges.

(a) \( \displaystyle \frac{(2k+1)(3k-1)}{(k+1)(k+2)^2} \) (Comparison Test, limit form)

(b) \( \displaystyle \frac{1}{k^{\frac{1}{k}}k} \) (Comparison Test, limit form)

(c) \( \displaystyle \frac{k!}{k^k} \) (Ratio Test)

(d) \( \displaystyle \binom{2k}{k}^{-1}(4-10^{-23})^k \) (Ratio Test)

(e) \( 2^{-k}k \) if \( k \) is of the form \( 2^m \) and \( 0 \) otherwise (Ratio Test)

For each of these series, write brief comments on possible alternative approaches.

Q2

(a) Use the Integral Test to prove that \( \displaystyle \sum_{k \geq 3} \frac{1}{k(\log k)^p} \) converges if \( p > 1 \) and diverges if \( 0 < p \leq 1 \).

(b) For which positive values of \( \alpha \) (\(\alpha \) not assumed to be a natural number) does \( \displaystyle \sum \frac{k^{-\alpha}}{1+\alpha^{-k}} \) converge?

Q3

For which of the following choices of \( a_k \) is  \( \sum a_k \) convergent? Justify your answers (using standard convergence tests where appropriate).

1. \( \displaystyle \frac{k + 2^k}{2^k k} \);
2. \( \displaystyle \frac{1}{k} \sin \frac{1}{k} \);
3. \( \displaystyle \frac{\sinh k}{2^k} \);
4. \( \displaystyle (-1)^{k-1}\frac{\log k}{\sqrt{k}} \);
5. \( \displaystyle \begin{cases} 1/k^2 & \textrm{ if $k$ is odd,} \\ -(\log k)/k^2 & \textrm{ if $k$ is even;}\end{cases} \)
6. \( \displaystyle \left(1 - \frac{1}{k}\right)^{2k} \).

Q4

For each of the following power series \( \sum c_k x^k \), establish which of the following is true: 

1. \( \sum |c_k x^k| \) converges for all \( x \in \mathbb{R} \); 
2. \( \sum |c_k x^k| \) converges only for \( x = 0 \); 
3. \( \sum |c_k x^k| \) converges for \( |x| < R \) and diverges for \( |x| > R \) for some \( R \in \mathbb{R}^{>0} \), which you should determine.

1. \( \displaystyle \sum k^{2020}x^k \);
2. \( \displaystyle \sum \frac{x^k}{2^k k^4} \);
3. \( \displaystyle \sum 2^k x^{k!} \);
4. \( \displaystyle \sum k^k x^k \);
5. \( \displaystyle \sum \frac{1}{(4k)!} x^{2k} \);
6. \( \displaystyle \sum \sin(k) x^k \).
   

What can you say in each case about the values of \( x \) for which \( \sum c_k x^k \) converges?

[The aim of this problem is to give you experience relating to the concept of the radius of a convergence of a power series.  The idea is that you use standard convergence tests for series to study these power series, not that you should use results from the last section of the course or elsewhere.]

Q5

Give either a justification or a counterexample for each of the following statements about real series.

1. If \( k a_k \to 0 \) as \( k \to \infty \) then \( \sum a_k \) converges.
2. If \( \displaystyle \lim_{k \to \infty} \frac{a_{k+1}}{a_k} \) exists and equals \( L \), where \( L > 1 \), then \( \sum a_k \) diverges.
3. If \( \sum a_k \) converges and \( \frac{a_k}{b_k} \to 1 \) then \( \sum b_k \) converges.

Q6

(Optional, or for use for consolidation later.)

(a) Prove that the series \( \displaystyle \sum_{k \geq 2}(\log k)^{-k} \) and \( \displaystyle \sum_{k \geq 2}(\log k)^{-\log k} \) converge.

(b) Prove that for any constant \( \alpha > 0 \), the series \( \displaystyle \sum_{k \geq 2}(\log k)^{-\alpha} \) diverges.