7.1 Sequences

Let \((a_ n)_{n∈ℕ}⊆ ℝ\) be a real-valued sequence (as defined in 74).

Given \(N∈ ℕ\) we will write \(\sup _{n≥ N} a_ n\) in the following, instead of \(\sup \{ a_ N,a_{N+1}\ldots \} \), and similarly for the infimum. (This is in accordance with 188)

E210

[0CP] Prerequisites:187.
We have that \(\sup _{n≥ N} a_ n=𝜎∈ \overlineℝ\) if and only if

\begin{eqnarray} & & ∀ n≥ N, a_ n≤ 𝜎 \label{eq:sup_ con_ ge_ L} \quad \text{e}\\ & & ∀ L{\lt} 𝜎, ∃ n≥ N, a_ n {\gt}L \label{eq:sup_ con_ le_ L} \end{eqnarray}

(note that if \(𝜎=∞\) the first is trivially true, while if \(𝜎=-∞\) the latter is true because there are no \(L\)).

Solution. 213

[0CQ]It follows from the characterization 183.

E210

[0CR] Let \((a_ n )_{n∈ℕ}\) be a sequence with \(a_ n ∼ n^ n\) . Prove that, setting \( s_ n{\stackrel{.}{=}}∑_{k=0}^ n a_ n\) we have \(s_ n ∼ a_ n\) .

E210

[0CS] Let \(e_ n,d_ n\) be two real sequences such that \(d_ n≤ e_ n\) for each \(n\), and suppose that \(\limsup _ n e_ n=\liminf _ n d_ n=b\) (possibly infinite): then show that \(\lim _ n e_ n=\lim _ n d_ n=b\). Hidden solution: [UNACCESSIBLE UUID ’0CT’]

E210

[0CV]Prerequisites:3,2.(Solved on 2022-11-24) Let \(a_ n,b_ n\) real valued sequences, show that

\[ \limsup _{n→∞} (a_ n+b_ n) ≤ (\limsup _{n→∞} a_ n) + (\limsup _{n→∞} b_ n)~ ~ ; \]

find a case where inequality is strict. Hidden solution: [UNACCESSIBLE UUID ’0CW’]

E210

[0CX] Difficulty:*.

Let \(a_{n,m}\) be a real valued sequence ¹ with two indexes \(n,m∈ℕ\). Suppose that

• for every \(m\) the limit \(\lim _{n→ ∞} a_{n,m}\) exists, and that

• the limit \(\lim _{m→ ∞} a_{n,m}=b_ n\) exists uniformly in \(n\) and is finite, i.e.

  \[ ∀ \varepsilon {\gt}0 ,~ ∃ m∈ℕ~ ∀ n∈ℕ ,~ ∀ h≥ m ~ ~ | a_{n,h}-b_ n|{\lt}\varepsilon ~ ~ . \]

then

\begin{equation} \lim _{n→ ∞} \lim _{m→ ∞} a_{n,m}= \lim _{m→ ∞} \lim _{n→ ∞} a_{n,m}\label{eq:limlimlimlim} \end{equation}

214

in the sense that if one of the two limits exists (possibly infinite), then the other also exists, and they are equal.

Find a simple example where the two limits in 214 are infinite.

Find an example where \(\lim _{m→ ∞} a_{n,m}=b_ n\) but the limit is not uniform and the previous equality 214 does not apply.

Hidden solution: [UNACCESSIBLE UUID ’0CZ’]

E210

[0D0]Prerequisites:5,3.Let again \(a_{n,m}\) be a real valued sequence with two indices \(n,m∈ℕ\); suppose that, for every \(n\), the limit \(\lim _{m→ ∞} a_{n,m}=b_ n\) exists, is finite and is uniform in \(n\); suppose that the limit \(\lim _ n b_ n\) exists and is finite. Can it be concluded that the limits \(\lim _{n→ ∞}a_{n,m}\) exist for each fixed \(m\)? Can we write an equality as in eqn. 214 in which, however, on the RHS we use the upper or lower limits of \(a_{n,m}\) for \(n→ ∞\), instead of the limits \(\lim _{n→ ∞}a_{n,m}\)?

Hidden solution: [UNACCESSIBLE UUID ’0D1’]

E210

[0D2]Difficulty:*. Show that from any sequence \((a_ n)_ n\) we can extract a monotonic subsequence. Hidden solution: [UNACCESSIBLE UUID ’0D3’]

E210

[0D4]Difficulty:*. Show that from any sequence \((a_ n)_ n⊆ ℝ\) we can extract a monotonic subsequence such that

\[ \lim _{k→∞}a_{n_ k}=\limsup _{n→∞} a_ n \quad . \]

Hidden solution: [UNACCESSIBLE UUID ’0D5’]

E210

[0D6]Topics:Euler-Mascheroni constant.Prerequisites:4.

Show that the limit

\[ 𝛾 = \lim _{n → ∞ } \left( ∑_{k=1}^ n \frac 1{k} - \log ( n) \right)\quad . \]

exists and is finite. This \(𝛾\) is called Costante di Eulero - Mascheroni. It can be defined in many different ways (see the previous link) including

\[ 𝛾 = ∫_ 1^∞\left(\frac{1}{⌊ x⌋}-\frac{1}{x}\right)\, {\mathbb {d}}x \]

where the parentheses \(⌊ ⋅ ⌋\) indicate the floor function \(⌊ x ⌋{\stackrel{.}{=}}\max \{ n∈ℤ :n≤ x\} \). In the image 1 the constant \(𝛾\) is the blue area.

Image by William Demchick, Creative Commons Attribution 3.0 Unported License, taken from wikipedia.

Hidden solution: [UNACCESSIBLE UUID ’0D8’]

E210

[0D9]

Let \(a_ k = \sqrt[3]{ k^ 3 + k} − k\). Prove that

\[ ∑_{k=1}^ n a_ k ∼ \frac{1}{3} \log (n) \]

that is, the ratio between the two above sequences tends to \(1\) when \(n → ∞\). Hidden solution: [UNACCESSIBLE UUID ’0DB’][UNACCESSIBLE UUID ’0DC’]

E210

[0DD]Note:Exercise 1 from the written exam 9 April 2011.Let \((a_ n)\) be a sequence of real numbers, with \(a_ n≥ 0\).

1. Show that if \( ∑_{n=1}^∞ a_ n\) converges then also

   \[ ∑_{n=1}^∞ a_ n^ 2 \quad \hbox{e} \quad ∑_{n=1}^∞ \left(a_ n ∑_{m=n+1}^∞ a_ m\right) \]

   converge.

2. Assuming moreover that \(∑_{n=1}^∞ a_ n\) is convergent, let’s define

   \[ a=∑_{n=1}^∞ a_ n ~ ~ ,~ ~ b=∑_{n=1}^∞ \left(a_ n ∑_{m=n+1}^∞ a_ m \right)~ ~ ,~ ~ c=∑_{n=1}^∞ a_ n^ 2 \]

   then show that \(a^ 2=2b+c\).

Summation by parts

E216

[217]Suppose \((a_ n)_ n,(b_ n)_ n\) are sequences of real numbers and \(c_ n\) is defined by 233; let then

\[ A_ n=∑_{h=0}^ n a_ h~ ~ ,~ ~ B_ n =∑_{h=0}^ n b_ h ~ ~ ,~ ~ C_ n=∑_{h=0}^ n c_ h \]

the partial sums of the three series; suppose that \(∑_{n=0}^∞ b_ n=B\) is convergent: show that

\[ C_ n=∑_{i=0}^ n a_{n-i}B_ i=∑_{i=0}^ n a_{n-i}(B_ i-B)+A_ nB \quad . \]

Hidden solution: [UNACCESSIBLE UUID ’216’]

E216

[21H]Note:Taken from Rudin [ 22 ] Prop. 3.41.

Let \((a_ n)_ n(b_ n)_ n,\) be sequences, let \(A_ n=∑_{k=0}^ n a_ k\) and \(A_{-1}=0\), \(0≤ p ≤ q\), then

\[ ∑_{n=p}^{q} a_ n b_ n = ∑_{n=p}^{q-1} A_ n (b_ n- b_{n+1}) + A_ q b_ q - A_{p-1} b_ p \quad . \]