Instructions

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Teamwork is encouraged but every person must turn in their own assignment. Please reference who you worked with and all external sources used. External help (not full solutions) or information is permitted but must be acknowledged. Lecture notes, OH help, course books, recorded lectures are not considered external and can be used without reference.

When solving a point of a problem you may give for granted any problem or point that appears prior to it, including problems on previous HW sheets (even if you didn't manage to solve them) but must referenced (e.g. "I use the proven fact that…" or "I use the fact from HW1P2 that…").

All statements require proof, unless specified otherwise.

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This is an EXTRA HW. If you do not turn it in you will not receive any grade penalty for it.

You will still have 1.5 worst HW dropped from HW00+ (HW01-HW08) with HW00 weighing 50% of a full homework.

If you decide to turn this HW in, you will have the worst 1.5 HW dropped from HW00+ (HW01-HW08)+ HW-extra with HW00 and HW-extra weighing 50% of a full homework.

1

Show Cauchy's condensation criterion.

Let $f : [0,+∞) → \R $ be positive and non-increasing.

Consider the quantities:

\begin{equation*} \begin{aligned} &f(1)+\int_{1}^{+\infty}f(x)\dd x \\ &\lim_{N}\sum_{k=0}^{N}f(2^{k})2^{k} \\ &\lim_{N}\sum_{k=1}^{N}f(k) \end{aligned} \end{equation*}

1.1

  • Show that any if any one of them is finite, then all of them are.
  • Show that they are all off by a factor i.e. find \(C\) independent of \(f\) such that
\begin{equation*} \mf{A}\leq C \mf{B} \end{equation*}

where \(\mf{A}\) and \(\mf{B}\) are any of the three quantities above.

1.2

Deduce that

\begin{equation*} \begin{aligned}[t] \lim_{N}\sum_{k=1}^{N} \frac{1}{k^s} \leq \infty \quad \iff \quad s \gt 1 \end{aligned} \end{equation*}

1.3

Suppose that \(a_k\) are real numbers such that \(|a_k|\leq C (1+|k|)^{n+1+s}\) for some \(C\gt 0\) some \(n\in\N\), and some \(s\gt 0\)

Show that

\begin{equation*} \lim_{N}\sum_{k\in\N} a_k \cos(2 \pi k x) \end{equation*}

converges in \(\Big(C^n_b(\R;\R),\|\cdot\|_{C^n} \Big)\).

2

Do HW06P3.2

3

Do HW03P2.6