Found problems: 51
Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$
for $n\ge2$.
Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\,
\frac{1}{F(2^n)}}$ is a rational number.
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.
Evaluate
$$\lim_{n\to \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^2 +j^2 }.$$
For each $x$ with $|x|<1$, compute the sum of the series
$$1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.$$
Calculate $ \sum_{i=2}^{\infty}\frac{i^2-2}{i!} . $
Given a positive integer $N$. Prove that
\[\sum_{m=1}^N \sum_{n=1}^N \frac{1}{mn^2+m^2n+2mn}<\frac{7}{4}.\]
[i]Proposed by tan-1[/i]
Which of the following series are convergent?
$\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$
$\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$
$\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$
$\textbf{(D)}~\text{None of the above}$
Let $(a_n )$ be a sequence of real numbers satisfying the inequalities
$$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$
and such that the series
$$\sum_{n=0}^{\infty} a_n $$
converges. Prove that
$$\lim_{n\to \infty} n a_n = 0.$$
For a real number $a$ and an integer $n(\geq 2)$, define
$$S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}$$
Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real.
Prove that the sequence $(a_n)$, where
$$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.
Let be $\gamma_1$ a circumference of radius $r$ and $P$ an exterior point that is at distance $a$ from the centre of $\gamma_1$. We build two tangent lines $r,s$ to $\gamma_1$ from $P$ and $\gamma_2$ is constructed as a smaller circumference, tangent to both lines and, also, tangent to $\gamma_1$. We construct inductively $\gamma_{n+1}$ as a tangent circumference to $\gamma_{n}$ and, also, tangent to $r$ and $s$. Determine:
a) The radius of $\gamma_2$.
b) The radius of $\gamma_n$.
c) The sum of the lengths of $\gamma_1, \gamma_2, \gamma_3, ...$.
Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that
$$\sum_{n=1}^{\infty} b_n$$
converges.
Let the series $$s(n,x)=\sum \limits_{k= 0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-nx)}{n!}$$ Find a real set on which this series is convergent, and then compute its sum. Find also $$\lim \limits_{(n,x)\to (\infty ,0)} s(n,x)$$
Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$
Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $
[b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $
[b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $
[i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $
[i]Cătălin Țigăeru[/i]
Consider the harmonic series $\sum_{n\ge1}\frac1n=1+\frac12+\frac13+\ldots$. Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms $\frac1k$.)
Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and
$$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$
for every $n\geq 1$.
(1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$.
(2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.
Find
(i) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i^{2}}}$.
(ii) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i}}$.
(iii) $\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}$.
[b]2.[/b] Show that the series
$\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}$
is convergent for every positive integer N and any real numbers a and b. [b](S. 25)[/b]
Assume that the complex numbers $a_1 , a_2, \ldots$ are all different from $0$, and that $|a_r - a_s| >1$ for $r\ne s.$
Show that the series
$$\sum_{n=1}^{\infty} \frac{1}{a_{n}^{3}}$$
converges.
Consider an infinite series whose $n$-th term is $\pm (1\slash n)$, the $\pm$ signs being determined according
to a pattern that repeats periodically in blocks of eight (there are $2^{8}$ possible patterns).
(a) Show that a sufficient condition for the series to be conditionally convergent is that there
are four "$+$" signs and four "$-$" signs in the block of eight signs.
(b) Is this sufficient condition also necessary?
Let $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions:
[list]
[*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$,
[*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$,
[*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$.
[/list]
Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$?
[i]Proposed by András Imolay, Budapest[/i]
Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.
For all positive integers $n,$ let $r_n$ be defined as \[r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.\]Prove that $\sum_{r=1}^\infty r_i=0.$
Given \( b > 0 \), consider the following matrix:
\[
B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix}
\]
Denote by \( e_i \) the top left entry of \( B^i \). Prove that the following limit exists and calculate its value:
\[
\lim_{i \to \infty} \sqrt[i]{e_i}.
\]