# Questions tagged [faq]

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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### 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 ...
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### 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 $(x,x) \not\in R$ ...
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### 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 ...
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### 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 ...
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### 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},\\ \...
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### 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 ...
1 vote
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### 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 ...
804 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?
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### 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 ...
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### 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 $n=k+1$...
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### 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 $(a^{-1})^m=e$ ...
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### 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?
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### 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.
1 vote
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### 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.
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### 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 ... 9k 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 ...
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### 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 ...
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### 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 ...