inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the divergence of the vector field \(r^n \vec{r}\), where \(r\) is the magnitude of the position vector \(\vec{r}\) and \(n\) is a constant, using tensor notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n + 3) r^n}
Determine the asymptotic behavior of the function \( \left(\frac{c+\sqrt{x}}{-c+\sqrt{x}}\right)^x \exp(-2c\sqrt{x}) \) as \( x \to \infty \), where \( c \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Estimate the number of digits of accuracy that can be expected in the solution \( x \) of the linear system \( Ax = b \), given that the entries of \( A \) have about 6 digits of accuracy, \( b \) has about 4 digits of accuracy, and the condition number of \( A \) is about \( 10^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the infimum of the normalized distances between the zeros of the Riemann zeta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine all values of \(x\) for which the power series \(\sum_{n=0}^\infty \frac{(x-2)^n}{4^n\sqrt{n}}\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 6)}
Evaluate the limit: \[ \lim_{n \to \infty} (-1)^n \sin\left(\pi(n^2+0.5n+1)^{\frac{1}{2}}\right)\sin\left(\frac{(n+1)\pi}{4n}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given a sequence of identically distributed continuous random variables $\xi_1, \xi_2, \ldots, \xi_n, \ldots$, where each variable has the same distribution, let $\nu$ denote the minimum index $k$ such that $\xi_{k-1} > \xi_k$. Find the expected value of $\nu$, denoted as $\mathbb{E}(\nu)$. \( \text{Express your answer in terms of known constants or functions.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
63
}
Determine the dual space of the sum $X=L^2(\mathbb{R})+L^3(\mathbb{R})$ using the norm defined in the problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L^2(\mathbb{R}) \cap L^{3/2}(\mathbb{R})}
Evaluate the integral \( \int_{\frac{3\pi}{2}}^{\frac{7\pi}{2}} \frac{e^{\tan^{-1}(\cos(x))}}{e^{\tan^{-1}(\sin(x))} + e^{\tan^{-1}(\cos(x))}} \,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Determine the number of partial orders on a set of 4 elements, given that the number of partial orders on a set of 3 elements is 19.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{219}
Find the inverse Z-transform of \( \frac{z}{z+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^n u[n]}
Determine the number of three-digit numbers such that:
- The number formed by the first two digits is prime.
- The number formed by the last two digits is prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
In an acute triangle \( ABC \), let \( M \) be the midpoint of \( AC \). A circle \( \omega \) passing through \( B \) and \( M \) intersects the sides \( AB \) and \( BC \) again at points \( P \) and \( Q \), respectively. Let \( T \) be the point such that the quadrilateral \( BPTQ \) forms a parallelogram. If \( T \) lies on the circumcircle of triangle \( ABC \), find all possible values of the ratio \( \frac{BT}{BM} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
A tetrahedron inscribed in a parallelepiped cuts off four tetrahedra from the parallelepiped with its face planes. What fraction of the volume of the parallelepiped is the volume of the tetrahedron whose vertices are the centroids of the cut-off tetrahedra?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{24}}
For four distinct integers, all pairwise sums and pairwise products were calculated. The obtained sums and products were written on the board. What is the smallest number of distinct numbers that could be on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Say that a 4-digit positive integer is *mixed* if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits.  For example, 2013 is mixed.  How many 4-digit positive integers are mixed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
Find all pairs of prime numbers whose difference of squares is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 2)}
A random walk is initiated at a vertex of a tetrahedron, with each of the three edges connected to this vertex being equally likely to be traversed. Determine the expected number of steps required to reach the opposite vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A computer company has a chance to get four different types of contracts: hardware, software, network services, and maintenance. The probability that the company will get the hardware contract is 3/4, the probability it will not get the software contract is 3/5, the probability of getting a network services contract is 2/3, and the probability of getting a maintenance contract is 1/2. If the probability of getting at least one contract is 7/8, what is the probability that the company will get all four contracts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/10}
On a sphere, there are four points A, B, C, and D. If AB, AC, and AD are mutually orthogonal and AB = AC = AD = 4, then calculate the surface area of the sphere. Express your answer in terms of pi (π).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\pi}
Given that $n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in a clockwise direction, and the sum of all numbers is 278, how many different values can $n$ take? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
某班准备购买班服,确定从A, B两种款式中选出一种统一购买，现在全班50位同学赞成购买A, B款式的人数分别为20,30位，为了尽量统一意见，准备在全班进行三轮宣传,每轮宣传从全班同学中随机选出一位，介绍他赞成款式的理由，假设每轮宣传后，赞成该同学所选款式的不会改变意见，不赞成该同学所选款式的会有5位改变意见，赞成该同学所选款式.
设经过三轮宣传后赞成A款式的人数为X，求随机变量X的期望.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{409}{25}$}
According to the rules of a chess match, the winner is the one who gains two victories over the opponent. Draws do not count. The probabilities of winning for the opponents are equal. The number of decisive games in such a match is a random variable. Find its expected value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given point $M(3,t)$, if there exist two points $A$ and $B$ on circle $O$: $x^{2}+y^{2}=6$ such that $\angle AMB=90^{\circ}$, then the range of $t$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \sqrt {3}\leqslant t\leqslant \sqrt {3}}
Let $\phi (x, u)$ be the smallest positive integer $n$ so that $2^u$ divides $x^n + 95$ if it exists, or $0$ if no such positive integer exists. Determine$ \sum_{i=0}^{255} \phi(i, 8)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2704}
A sum of Rs. 5060 is divided among 4 persons, p, q, r, and s, in such a way that the ratios p : q : r : s = 3 : 5 : 7 : 9 and q : r : s = 10 : 11 : 13. Find the amount that person r receives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1475.83}
The function $f(x) = x^n + 1$ always passes through a fixed point. What are the coordinates of this fixed point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240 d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Find the smallest positive integer \( n \) such that, if there are initially \( 2n \) townspeople and 1 goon, then the probability that the townspeople win is greater than 50%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
The numbers from 1 to 8 are placed at the vertices of a cube so that the sum of the numbers at any three vertices, which lie on the same face, is at least 10. What is the minimum possible sum of the numbers on one face?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
For which integers \( n \) do there exist integers \( a \) and \( b \) such that \( n + a^{2} = b^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \not\equiv 2 \pmod{4}}
There are nine reduced quadratic polynomials written on a board: \(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}\). It is known that the sequences \(a_{1}, a_{2}, \ldots, a_{9}\) and \(b_{1}, b_{2}, \ldots, b_{9}\) are arithmetic progressions. It turns out that the sum of all nine polynomials has at least one root. What is the maximum number of the original polynomials that can have no roots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has a diameter of 8 cm. In Crate A, the pipes are packed directly on top of each other in 25 rows of 8 pipes each across the width of the crate. In Crate B, pipes are packed in a staggered (hexagonal) pattern that results in 24 rows, with the rows alternating between 7 and 8 pipes. 

After the crates have been packed with an equal number of 200 pipes each, what is the positive difference in the total heights (in cm) of the two packings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200 - 96\sqrt{3}}
Consider a sequence $x_{n}$ such that $x_{1}=x_{2}=1, x_{3}=\frac{2}{3}$. Suppose that $x_{n}=\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}}$ for all $n \geq 4$. Find the least $n$ such that $x_{n} \leq \frac{1}{10^{6}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
How many people are needed to be certain that at least two of them have the same birthday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{366}
A convex hexagon has interior angles with measures $2x+2$, $3x$, $4x+3$, $5x$, $6x-1$, and $7x$. What is the measure of the largest angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5012}{27}\text{ degrees}}
Arun, Tarun, and Chun can do a work in 10 days. Arun can do the work alone in 60 days, and Tarun can do the work alone in 40 days. After working together for 4 days, Tarun leaves for his village, and Chun, who can finish the work alone in 20 days, joins. How many days are required for Arun and Chun to complete the remaining work together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \( a, x, y \) be three positive integers greater than 100, satisfying the equation \( y^{2} - 1 = a^{2}(x^{2} - 1) \). Find the minimum value of \( \frac{a}{x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the probability that \( G(x,y,z) > 100 \), where \( G(x,y,z) = 20000 - 800x - 100000y + 200000z \). The variables \( x, y, z \) are independent and normally distributed with the following parameters: \( x \sim N(150, 20^2) \), \( y \sim N(2, 0.8^2) \), and \( z \sim N(0.2, 0.02^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0007}
Evaluate the Ackermann function at the point (4, 0) to find the value of \( \operatorname{ack}(4, 0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Two persons start running simultaneously around a circular track of a certain length from the same point at speeds of 20 kmph and 40 kmph. They meet for the first time anywhere on the track after 14.998800095992321 seconds when they are moving in the opposite direction. What is the length of the track?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{249.98}
Compute
\[\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{15} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{16} \rfloor}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{8}}
Five super-intelligent and vindictive pirates, ranked by height, must distribute 500 gold coins. The shortest pirate proposes a distribution, and if at least half of the pirates, including the proposer, agree, the distribution is accepted. Otherwise, the proposer is eliminated, and the next shortest proposes. What is the optimal number of coins the shortest pirate can receive in a Nash Equilibrium?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{498}
A room is illuminated by two light bulbs. The bulbs are replaced only when both have burned out. The lifetimes of the bulbs are independent and identically distributed, following an exponential distribution with parameter \( \lambda = 1 \). What fraction of the time is the room only half-lit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Let $X_k$, $k \geq 0$, be iid random variables with pmf $p_j = P(X_1 = j)$ for $1 \leq j \leq m$, where $\sum_{j = 1}^{m}p_j = 1$. Define $T = \min\{n > 0 : X_n = X_0\}$ as the first time we observe the initial value again. Compute $E[T]$. \(E[T]\) is the expected value of the first time the random variable $X_n$ equals $X_0$ again.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Find the value of \( k \) such that \[ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + k \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x = 0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
John ends up serving on jury duty for a case involving three defendants with different roles in the crime. Jury selection takes 2 days. The trial itself lasts 4 times as long as jury selection. However, on each day of the trial, 2 hours are spent on reviewing evidence and 1 hour for lunch, apart from the regular 6 hours daily trial session. It is a complicated trial. 

The jury must deliberate separately for each defendant. The number of hours spent in jury deliberation for the first defendant was the equivalent of 6 full days. For the second defendant, the jury took 4 full days of deliberation. The third defendant required 5 full days of deliberation. They spend 14 hours a day in deliberation, considering a 2-hour break for lunch and rest in between. 

How many days does John spend on jury duty in total, taking into account the extra time spent on evidence review, lunch, and breaks during the trial and deliberation periods for all three defendants?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Evaluate the integral: $$\int _{0}^{1} \frac{7x^4+5x^2+4x}{\sqrt {x^3+x+1}} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
Evaluate the limit \( \lim_{n\rightarrow\infty} \sum^{n}_{k=0}\left|\frac{2\pi\cos(k\pi(3-\sqrt{5}))}{n}\right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the asymptotic behavior of the integral \((n+1)\int_{0}^{1}x^n\ln(x+1)dx\) as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Calculate the flux of the vector field \( v = (3y_1 - 2y_2^2 + y_3^2, 3y_1^2 - 2y_2 + y_3^2, 3y_1^2 - 2y_2^2 + y_3) \) through the surface \( \tilde{Y} \), where \( \tilde{Y} \) is the image of the parameterization \( \varphi: K \to \mathbb{R}^3 \) with \( K = [0, 2\pi] \times [0, 1] \) and \( \varphi(x) = ((\cos x_1)(x_2 - x_2^3), (\sin x_1)(x_2 - x_2^3), x_2^2 + \frac{1}{3} x_2^6) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{5}}
Find the residue of the function \( f(z) = \frac{1}{z^2 \sin(z)} \) at the singularity \( z=0 \) for \( |z| < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Let \( G = S_5 \) be the symmetric group on 5 elements. Let \( H = \langle (1,2,3) \rangle \) be a subgroup of \( G \). Determine the number of subgroups of \( G \) that are conjugate to \( H \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a duel involving three participants, A, B, and C, each takes turns shooting at the next person in the sequence: A shoots at B, B shoots at C, and C shoots at A. The duel continues in this cycle until only one participant remains. A and C have a probability of 0.5 of hitting their target, while B has a probability of 0.75. All shots are independent. What is the probability that A is the sole survivor of the duel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
Find a closed form expression in terms of known constants for the sum of the following integrals, which converges to approximately 0.1746809:
\[-\frac{1}{2}\int _0^{1}\frac{\left(\frac{1-t}{1+t}\right)\arctan \left(t\right)}{t\left(\ln t\right)^2}dt-\frac{\pi }{4}\int _0^{1}\frac{1}{\ln \left(t\right)\left(1+t\right)^2}dt\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{18}}
In a regular nonagon (9-sided polygon) inscribed in a circle, let AB and BC be two adjacent sides, and let M be the midpoint of AB. Let N be the midpoint of the radius OT, where OT is perpendicular to BC. Find the measure, in degrees, of the angle OMN.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Determine the total number of different game situations possible in a standard Connect Four game played on a 7-column by 6-row grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4531985219092}
Let \( V \) be the vector space of all \( 3 \times 3 \) matrices, and let \( A \) be the diagonal matrix:

\[ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Compute the determinant of the linear transformation \( T(X) = \frac{1}{2} (AX + XA) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{8}}
Given a non-singular $n \times n$ matrix $A$ with all real eigenvalues, such that $\text{Tr}(A^2) = \text{Tr}(A^3) = \text{Tr}(A^4)$, find the value of $\text{Tr}(A)$. Here, $\text{Tr}(X)$ denotes the trace of matrix $X$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
The "Rhind Mathematical Papyrus" is one of the oldest mathematical texts in the world. It contains the following problem: Divide 100 loaves of bread among 5 people so that each person receives an amount forming an arithmetic sequence, and the sum of the larger three portions is $\dfrac{1}{7}$ of the sum of the smaller two portions. What is the largest portion?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{115}{3}}
Find the minimum value of \( f(a, \theta) = \cos^3 \theta + \frac{4}{3a \cos^2 \theta - a^3} \) given that \( 0 < a < \sqrt{3} \cos \theta \) and \( \theta \in \left[-\frac{\pi}{4}, \frac{\pi}{3}\right] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $T$ be a diagonalizable linear operator on a finite-dimensional vector space $V$ over a field $F$. Suppose the set $\{id_V, T, T^2, \ldots, T^d\}$ is linearly independent and $T$ has $r$ distinct eigenvalues. If $q(T)=0$ for the polynomial $q(T)=a_0+a_1T+a_2T^2+\cdots a_dT^d$ where $a_i\in F$ are not all zero, determine the values of $q(c_i)$, where $c_i$ are the distinct eigenvalues of $T$. Provide your answer as a set of equations $q(c_i)=0$ for each $c_i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q(c_i) = 0}
Calculate the residue of the function \( \frac{(z^6-1)^2}{z^5(2z^4-5z^2+2)} \) at the pole \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{8}}
Calculate the angle between the face $P_0P_2P_3$ and the face $P_1P_2P_3$ of a pyramid with vertices at $P_0=(1,2,3)$, $P_1=(4,5,6)$, $P_2=(7,-4,9)$, and $P_3=(10,1,8)$. Express your answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.2}
Determine the minimum value of the polynomial \( P(x) = (x-x_i)(x-x_{i+1})(x-x_{i+2})(x-x_{i+3}) \) in the interval \((x_{i+1}, x_{i+2})\), given that \(x_j - x_{j-1} = h\), without using calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the Laplace transform of the derivative of the Dirac delta function shifted by 1 unit, \( \delta'(t-1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s e^{-s}}
On a table, there are 4 glasses in a row: the first and third contain orange juice, and the second and fourth are empty. Valya randomly pours juice from one full glass to one of the two empty glasses. What is the expected number of pourings needed for the first time when the first and third glasses are empty, and the second and fourth glasses are full?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the Riemann-Stieltjes integral: \( \int_{-1.2}^{3.9} x d[x] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Provide an example of a set with cardinality \(\aleph_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega_2}
Evaluate the sum of the series \( \sum_{n=-\infty}^{\infty}\frac{1}{n^2+1} \) using the Poisson summation formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \coth(\pi)}
A circular sector is rotated about one of its boundary radii, forming a solid. The spherical surface area of this solid is equal to its conical surface area. Determine the sine of the central angle of the circular sector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{5}}