7.3 Series

Tests

Exercises

E228

[214](Solved on 2022-12-13) Let \(𝛼{\gt}0\); use Raabe’s criterion 228 to study the convergence of the series

\[ ∑_{n=1}^∞ \frac{1}{n^𝛼} \]

Hidden solution: [UNACCESSIBLE UUID ’215’]

E228

[23D](Solved on 2022-12-13) Let \(𝛼{\gt}0\); use the condensation criterion 224 to study the convergence of the series

\[ ∑_{n=1}^∞ \frac{1}{n^𝛼} \]

E228

[0DW] Given a series \(∑_ n^∞ a_ n\) tell if the following conditions are necessary and/or sufficient for convergence.

\begin{eqnarray} ∀\varepsilon {\gt}0~ ∃ m∈{\mathbb {N}}~ ∀ n{\gt}m ~ ∀ k∈{\mathbb {N}}~ ~ \left|∑_{j=n}^{n+k} a_ k\right|{\lt}\varepsilon \\ ∀\varepsilon {\gt}0~ ∀ k∈{\mathbb {N}}~ ∃ m ∈{\mathbb {N}}~ ∀ n{\gt}m ~ \left|∑_{j=n}^{n+k} a_ k\right|{\lt}\varepsilon \\ ∀\varepsilon {\gt}0~ ∃ m∈{\mathbb {N}}~ ∀ n{\gt}m∀ k∈{\mathbb {N}}~ ~ ∑_{j=n}^{n+k} |a_ k| {\lt}\varepsilon \\ ∀\varepsilon {\gt}0~ ∀ k∈{\mathbb {N}}~ ∃ m∈{\mathbb {N}}~ ∀ n{\gt}m ~ ∑_{j=n}^{n+k} |a_ k|{\lt}\varepsilon \end{eqnarray}

Hidden solution: [UNACCESSIBLE UUID ’0DX’]

E228

[0DY](Proposed on 2022-12-13) Find two sequences \((a_ n)_ n,(b_ n)_ n\) with \(a_ n,b_ n{\gt}0\) such that \(∑_{n=0}^∞ (-1)^ n a_ n\) is convergent, \(∑_{n=0}^∞ (-1)^ n b_ n\) is non-convergent, and \(\lim _{n→∞} a_ n/b_ n=1\). Hidden solution: [UNACCESSIBLE UUID ’0DZ’]

E228

[0F0]Note:Exam of 9th APr 2011.Let \((a_ n)\) be a sequence of real numbers (not necessarily positive) such that the series \(∑_{n=1}^∞ a_ n\) converges to \(a∈{\mathbb {R}}\); let \(b_ n=\frac{a_ 1+\cdots +a_ n}{n}\); show that if the series \(∑_{n=1}^∞ b_ n\) converges then \(a=0\).

E228

[0F2](Proposed on 2022-12) Find two examples of \(a_{i,j}:{\mathbb {N}}× {\mathbb {N}}→ {\mathbb {R}}\)

• such that, for each \(i\), \(∑_ j a_{i,j} =0\), while for each \(j\), \(∑_ i a_{i,j} =∞\);

• such that, for each \(i\), \(∑_ j a_{i,j} =0\), while for each \(j\), \(∑_ i a_{i,j} =1\).

Can you find examples where moreover we have that \(|a_{i,j}|≤ 1\) for every \(i,j\)? [UNACCESSIBLE UUID ’0F3’]

Prove that

1. if \(∏_{n=0}^∞ a_ n\) converges then \(\lim _{n→+∞}a_ n=1\);

2. if the series \(∑_{n=0}^∞|a_ n-1|\) converges, then it also converges \(∏_{n=0}^∞ a_ n\);

3. find an example where the series \(∑_{n=0}^∞(a_ n-1)\) converges but \(∏_{n=0}^∞ a_ n=0\).

[0F5]We indicate with \({\mathcal P}_{\mathfrak f}({\mathbb {N}})\) the set of subsets \(B⊆ {\mathbb {N}}\) which are finite sets. This is said the set of finite parts.

We abbreviate \({\mathcal P}={\mathcal P}_{\mathfrak f}({\mathbb {N}})\) in the following.

Given a sequence \((a_ n)_ n\) of real numbers and a \(B∈{\mathcal P}\) we indicate with \(s(B)=∑_{n∈ B} a_ n\) the finite sum with indices in \(B\).

Suppose the series \(∑_{n=0}^∞ a_ n\) converge but not converge at all. Then:

• \(\{ s(F ) : F ∈{\mathcal P}\} \) it is dense in \({\mathbb {R}}\).

• There is a reordering \(σ\) of \({\mathbb {N}}\), that is, a bijective function \(σ:{\mathbb {N}}→{\mathbb {N}}\), such that all partial sums \(∑_{n=0}^ N a_{σ(n)}\) (at the variation of \(N\)) is dense in \({\mathbb {R}}\).

[UNACCESSIBLE UUID ’0F6’] [0F7]Note:This result is attributed to Riemann , see 3.54 in [ 22 ] ..

Let be given a sequence \((a_ n)_ n\) of real numbers such that \(∑_{n=0}^∞ a_ n\) converges (to a finite value) but \(∑_{n=0}^∞ |a_ n|=∞\); for each \(l,L\) with \(-∞ ≤ l ≤ L ≤ +∞\) there is a permutation \(𝜋:{\mathbb {N}}→{\mathbb {N}}\) such that, defining \(S_ N=∑_{k=0}^ N a_{𝜋(k)}\), we have that

[0F8]A sequence is given \((a_ n)_{n∈ {\mathbb {N}}}\) of positive real numbers such that \(\lim _{n→∞} a_ n=0\) and \(∑_{n=0}^∞ a_ n=∞\): prove that for every \(l ∈ {\mathbb {R}}\) there is a sequence \((ε_ n )_{n∈{\mathbb {N}}}\) with \(ε_ n ∈\{ 1,-1\} \) for each n, such that

If instead \(∑_{n=0}^∞ a_ n=S{\lt}∞\), what can be said about the set \(E\) of the sums \(∑_{n=0}^∞ (ε_ n a_ n)=l\), for all possible choices of \((ε_ n )_{n∈{\mathbb {N}}}\) with \(ε_ n ∈\{ 1,-1\} \) for every n?

• Analyze cases where \(a_ n=2^{-n}\) or \(a_ n=3^{-n}\)

• Show that \(E\) is always closed.

• Under what assumptions do you have that \(E=[-S,S]\)?

Hint. Let \(\tilde E\) be the set of sums \(∑_ n (ε_ n a_ n)=l\), to vary by \((ε_ n )_{n∈{\mathbb {N}}}\) with \(ε_ n ∈ \{ 0,1\} \) for each n; note that \(\tilde E=\{ (S+x)/2 : x∈ E\} \). [0F9]Note:Written exam of 12th Jan 2019.

Show that the following series converges

Hidden solution: [UNACCESSIBLE UUID ’0FB’] [21M](Proposed on 2022-12) Say for which \(𝛼{\gt}0,𝛽{\gt}0,𝛾{\gt}0\) you have that

converges. [23F](Proposed on 2022-12-13) Note:Written exam 29th January 2021.Let it be \(𝛼{\gt}0\). Say (justifying) for which \(𝛼\) the following series converge or diverge

• \[ ∑_{n=1}^∞ \left({\sqrt[4]{n^ 8+n^𝛼} - n^ 2 }\right) \]
• \[ ∑_{n=2}^∞ \left( \frac{1}{n^𝛼} - \frac{1}{n^𝛼+1} \right) \]
• \[ ∑_{n=2}^∞ \frac{1}{(\log _ 2 n) ^{𝛼\log _ 2(n)}} \]

  where the logarithms are in base 2.

Hidden solution: [UNACCESSIBLE UUID ’23G’] [20Z]Note:Task of 26 Jan 2016. (Solved on 2022-01-20)

Let

Show that \(\lim _{n→∞} z_ n=0\) but

Hidden solution: [UNACCESSIBLE UUID ’213’] [210] Note:exercise 2, written exam 15 January 2014. Let \((a_ n)_{ n ≥ 0}\) be a sequence of positive real numbers. Having defined \(s_ n =∑_{i=0}^ n a_ i \) prove that:

• the series \(∑_{n=0}^∞ a_ n\) converges if and only if the series \(∑_{n=0}^∞ a_{n}/s_ n\) converges;

• the series \(∑_{n=0}^∞ a_ n / (s_ n)^ 2\) converges.

Hidden solution: [UNACCESSIBLE UUID ’21K’]

[UNACCESSIBLE UUID ’0DT’] [UNACCESSIBLE UUID ’0DV’] [UNACCESSIBLE UUID ’0FC’] [UNACCESSIBLE UUID ’0FD’] [UNACCESSIBLE UUID ’0FF’] [UNACCESSIBLE UUID ’0FG’] [UNACCESSIBLE UUID ’0DH’] See also exercise 1.

Cauchy product

E233

[0FJ]If \(\forall n\in {\mathbb {N}}, a_ n,b_ n≥ 0\) show that

\[ ∑_{n=0}^∞ c_ n =∑_{n=0}^∞ a_ n ∑_{n=0}^∞ b_ n \]

with the convention that \(0⋅ ∞=0\).

E233

[0FK](Proposed on 2022-12-13) If the series \(∑_{n=0}^∞ a_ n\) and \(∑_{n=0}^∞ b_ n\) converge absolutely, show that the series \(∑_{n=0}^∞ c_ n\) converges absolutely and

\[ ∑_{n=0}^∞ c_ n =∑_{n=0}^∞ a_ n ∑_{n=0}^∞ b_ n\quad . \]

E233

[0FM] Prerequisites:1.Note:Known as: Mertens’ theorem..

If the series \(∑_{n=0}^∞ a_ n\) converges absolutely and \(∑_{n=0}^∞ b_ n\) converges, show that the series \(∑_{n=0}^∞ c_ n\) converges and

\[ ∑_{n=0}^∞ c_ n =∑_{n=0}^∞ a_ n ∑_{n=0}^∞ b_ n\quad . \]

Hidden solution: [UNACCESSIBLE UUID ’0FN’] [UNACCESSIBLE UUID ’218’]

See also exercise 1.