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This is meant for questions which are generalized forms of questions which get asked frequently. See tag details for more information.

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2 answers
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Overview of basic results in Stochastic Calculus

Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus? These can be in the form of resources (preferably accessible online) as well as directly writing out ...
• 4,331
0 votes
1 answer
162 views

Interpreting the row echelon form of an (augmented) matrix

Once we perform row reduction on a matrix, to put it in row echelon form, what does this tell us? How should we interpret the results? This question is intended as an FAQ. While plenty of questions ...
• 51.8k
33 votes
3 answers
2k views

Are there infinitely many primes of the form [X]? We probably don't know.

Are there infinitely many primes of the form [expression]? (We probably don't know. Sorry.) This question appears pretty often, with any number of various expressions. The sad reality is that the ...
• 3,049
9 votes
2 answers
2k views

Is there any variation known to the sum of two squares theorem?

Originally posed by Fermat and subsequently generalized as sum of two squares theorem, we can see the following statement. An integer greater than one can be written as a sum of two squares if and ...
• 362
3 votes
1 answer
2k views

How to solve linear recurrence relations with constant coefficients.

As questions regarding sequences that verifies a linear recurrence relation with constant coefficients are posted very often on this site and that there appear to be no reference post about it, so I ...
• 28.9k
2 votes
2 answers
5k views

Examples and Counterexamples of Relations which Satisfy Certain Properties

Definition: Given a set $X$, a relation $R$ on $X$ is any subset of $X\times X$. A relation $R$ on $X$ is said to be reflexive if $(x,x) \in R$ for all $x \in X$, irreflexive if $(x,x) \not\in R$ ...
• 30.2k
2 votes
5 answers
4k views

Proving that among any $2n - 1$ integers, there's always a subset of $n$ which sum to a multiple of $n$

How can one prove that among any $2n - 1$ integers, there's always a subset of $n$ which sum to a multiple of $n$? It is not hard to see this is equivalent to show that among $2n-1$ residue classes ...
• 49.7k
7 votes
3 answers
2k views

How to solve homogeneous linear recurrence relations with constant coefficients?

Consider a sequence $(a_n)_{n\in\mathbb N}$ defined by $k$ initial values $(a_1,\dots,a_k)$ and $$a_{n+k}=c_{k-1}a_{n+k-1}+\dots+c_0a_n$$ for all $n\in\mathbb N$. What are some ways to get closed ...
2 votes
0 answers
108 views

• 25.2k
22 votes
9 answers
4k views

Summation Theorem how to get formula for exponent greater than 3

I'm studying in the summer for calculus 2 in the fall and I'm reading about summation. I'm given these formulas: \begin{align*} \sum_{i=1}^n 1 &= n, \\ \sum_{i=1}^n i &= \frac{n(n+1)}{2},\\ \...
• 381
11 votes
1 answer
3k views

Volume of the intersection of two cylinders

I have two infinite cylinders of unit radius in $\mathbb{R}^3$, whose axes are skew lines. Say that the axis of one is centered on the $x$-axis, and the axis of the other is determined by the two ...
• 30.4k
1 vote
2 answers
13k views

Find $11^{644} \mod 645$ [duplicate]

Can someone just explain to me the basic process of what is going on here? I understand everything until we start adding 1's then after that it all goes to hell. I just need some guidance. The Problem ...
• 1,641
4 votes
5 answers
1k views

Proof of convergence of a recursive sequence

How do I prove that $x_{n+2}=\frac{1}{2} \cdot (x_n + x_{n+1})$ $x_1=1$ $x_2=2$ is convergent?
• 131
1 vote
1 answer
3k views

Volume of Intersection of cylinders (different radii)

I want to derive a formula for the area of the intersection of two rigth cylinders with different radii. To get an idea I attached a sketch. My idea is to determine the borders of $x$ and $z$ in ...
• 310
2 votes
6 answers
810 views

Prove by induction that $n^2<n!$

How can I show that $n^2<n!$ for all $n\geq 4$ Step 1 For $n=1$, the LHS=$4^2=16$ and RHS=$4!=24$. So LHS$<$ RHS. Step 2 Suppose the result be true for $n=k$ i.e., $k^2<k!$ Step 3 For $n=k+1$...
• 21
17 votes
4 answers
42k views

An element of a group has the same order as its inverse

If $a$ is a group element, prove that $a$ and $a^{-1}$ have the same order. I tried doing this by contradiction. Assume $|a|\neq|a^{-1}|$. Let $a^n=e$ for some $n\in \mathbb{Z}$ and $(a^{-1})^m=e$ ...
4 votes
4 answers
5k views

How to compute $3^{2003}\pmod {99}$ by hand? [duplicate]

Compute $3^{2003}\pmod {99}$ by hand? It can be computed easily by evaluating $3^{2003}$, but it sounds stupid. Is there a way to compute it by hand?
• 771
3 votes
2 answers
815 views

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
• 835
1 vote
1 answer
71 views

Will this series converge? If so, what is its limit?

If $a_n=(a_{n-1}+a_{n-2})/2$ and $a_1, a_2$ are given, will this series converge? And if so, what is the limit? By intuition I think it converges to $(a_1+2a_2)/3$ , but I am not able to prove it.
• 3,688
3 votes
4 answers
926 views

Calculating volume enclosed using triple integral

Calculate the volume enclosed between $x^2 + y^2=1$, $y^2+z^2=1$, $x^2+z^2=1$ I am supposed to solve this question using multiple integral. I am not able to visualize the resultant figure of whose I ...
33 votes
5 answers
57k views

Counting bounded integer solutions to $\sum_ia_ix_i\leqq n$

I want to find the number of nonnegative integer solutions to $$x_1+x_2+x_3+x_4=22$$ which is also the number of combinations with replacement of $22$ items in $4$ types. How do I apply stars and bars ...
74 votes
2 answers
12k views

Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Characteristic polynomial of a matrix ...
• 116k
11 votes
1 answer
8k views

Single Variable Calculus Reference Recommendations

This question is a generalization of the common question asking for calculus references. It is here to abstract away the repetition, and give a canonical resource for calculus references. I'm ...
10 votes
3 answers
18k views

• 231
7 votes
9 answers
10k views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without any ...
• 6,590
17 votes
7 answers
67k views

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer)

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer) I saw no example of this problem anywhere.. I saw an ...
• 1,415
158 votes
1 answer
35k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...