On a sequence

Consider the sequence of real numbers $\{x_n\}_{n=1}^{\infty}$ such that:

$$\lim_{n \rightarrow +\infty} \frac{x_1^{2p}+x_2^{2p}+\cdots+x_n^{2p}}{n}=0$$.

where $p$ is a positive integer. Prove that $\lim \limits_{n \rightarrow +\infty} \frac{x_1+x_2+\cdots+x_n}{n}=0$.

Does the converse hold?

Solution

Convergence of sequence (Euler - Mascheroni constant)

Let $\mathcal{H}_n$ denote the $n$-th harmonic number. Prove that the sequence:

$$\gamma_n = \mathcal{H}_n - \ln n$$

converges.

Solution

Sequence of zero limit

Let $\{a_n\}_{n \in \mathbb{N}}$ be a sequence of real number such that the series $\sum \limits_{n=1}^{\infty} \frac{a_n}{n}$ converges. Prove that:

$$\lim_{n \rightarrow +\infty} \frac{a_1+a_2+\cdots +a_n}{n}=0$$

Solution

Nested sequence and series

Let $a_n =\underbrace{\sin \left ( \sin \cdots \left ( \sin x \right )\cdots \right )}_{n \;\; {\rm times}}$. Examine if the series $\sum \limits_{n=1}^{\infty} a_n$ converges.

Solution

Sequence of series

Let $f_n: \mathbb{R} \rightarrow \mathbb{R}$ be defined as:

$$f_n(x)=\frac{\cos nx}{n^2}, \; \; n \in \mathbb{N}$$

Prove that the series $\displaystyle \sum_{n=1}^{\infty} f_n$ converges uniformly to a function $f:\mathbb{R} \rightarrow \mathbb{R}$ and that:

$$\int_0^{\pi/2} f(t)\, {\rm d}t = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^3}$$

Solution

Value of function

Let $f$ be a function defined as $f(1)=2$ and for every positive integer $n>1$ holds:

$$f(1)+f(2)+f(3)+\cdots +f(n) = n^2 f(n)$$

Evaluate the value $f(2013)$.

Source: CRUX

Solution

Limit of a sequence

Evaluate the limit of the sequence: $a_n = \sin \left( 2\pi \sqrt{n^2+n} \right)$.

Solution

Convergece of a sequence

Examine the convergence of the sequence:

$$\gamma_n=\left ( 1+a \right )\left ( 1+2a^2 \right )\left ( 1+3a^3 \right )\cdots\left ( 1+na^n \right)$$

for the different values of $a \in \mathbb{R}$.

Solution

Limit of a sequence

Find the limit of the sequence:

$$a_n= \left ( 1- \frac{1}{2^2} \right )\left ( 1- \frac{1}{3^2} \right )\cdots \left ( 1- \frac{1}{n^2} \right ), \;\; n \geq 2$$

Solution

The sequence takes its equivalent form:

$$\begin{aligned}
\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ) &=\prod_{k=2}^{n}\left [ \left ( 1- \frac{1}{k} \right )\left ( 1+ \frac{1}{k} \right ) \right ] \\
 &= \left [ \left (  1- \frac{1}{2} \right )\left ( 1- \frac{1}{3} \right ) \cdots \left ( 1- \frac{1}{n} \right ) \cdot \left ( 1+ \frac{1}{2} \right )\left ( 1+ \frac{1}{3} \right )\cdots \left ( 1+ \frac{1}{n} \right )  \right ] \\
 &=\frac{1}{2}\cdot \frac{2}{3}\cdots \frac{n-1}{n}\cdot \frac{3}{2}\cdot \frac{4}{3}\cdots \frac{n+1}{n} \\
 &=\frac{1}{2}\cdot \frac{n+1}{n} \xrightarrow{n \rightarrow +\infty}\frac{1}{2}
\end{aligned}$$

Hence the limit of the sequence is $1/2$.

Sequence

Let $0<a<b$ be real numbers. Define:

$$x_1=a, \;\; x_2=b, \;\; x_{2n+1}=\sqrt{x_{2n}x_{2n-1}}, \;\;x_{2n+2}= \frac{x_{2n}+x_{2n-1}}{2}$$

Prove that the sequence converges and find its limit.

Solution:

Recurrent sequence

Let $a_n$ be a real sequence such that $a_1=2$ and:
$$a_n=\frac{n+1}{n-1}\sum_{i=1}^{n-1}a_i, \;\; n  \geq 2$$

Express $a_n$ in an inductive form.

Solution