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Questions tagged [faq]

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

When you add a question marked faq, please also update the list of questions: List of Generalizations of Common Questions

The question which prompted this: Coping with *abstract* duplicate questions.

Note: Even though one might argue that tagging a question as faq should be enough, and there is no need to update the above list, updating the above list will serve to bring this policy back to attention and help raise awareness periodically.

112 questions
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2 answers

Significance of $\sqrt[n]{a^n} $?!

There is a formula given in my module: $$ \sqrt[n]{a^n} = \begin{cases} \, a &\text{ if $n$ is odd } \\ |a| &\text{ if $n$ is even } \end{cases} $$ I don't really understand the differences between them, kindly explain with an example.
Quixotic
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29
votes
5 answers

Units and Nilpotents

If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit. I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 = (u+a)x$. After tried several elements and…
Shannon
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27
votes
10 answers

How to determine equation for $\sum_{k=1}^n k^3$

How do you find an algebraic formula for $\sum_{k=1}^n k^3$? I am able to find one for $\sum_{k=1}^n k^2$, but not $k^3$. Any hints would be appreciated.
AlexHeuman
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27
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5 answers

Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?

I was wondering if there is a name for a function that satisfies the conditions $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$? Thanks and regards!
Tim
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25
votes
1 answer

Number of permutations of $n$ elements where no number $i$ is in position $i$

I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is not because 5 is in position 5. I know that the…
Will specht
24
votes
3 answers

Is $x^x=y$ solvable for $x$?

Given that $x^x = y$; and given some value for $y$ is there a way to expressly solve that equation for $x$?
Josh
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22
votes
2 answers

How to tell whether two graphs are isomorphic?

Suppose that we are given two graphs on a relatively small number of vertices. Here's an example: Are these two graphs isomorphic? (That is: is there a bijection $f$ from $\{A,B,\dots,I\}$, the vertex set of the first graph, to $\{1,2,\dots,9\}$,…
Misha Lavrov
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21
votes
1 answer

$\arcsin$ written as $\sin^{-1}(x)$

I know that different people follow different conventions, but whenever I see $\arcsin(x)$ written as $\sin^{-1}(x)$, I find myself thinking it wrong, since $\sin^{-1}(x)$ should be $\csc(x)$, and not possibly confused with another function. Does…
bobobobo
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21
votes
5 answers

How to compute the determinant of a tridiagonal matrix with constant diagonals?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & 0 \\ 0 & 2 & 5 & 2 & 0 & \cdots & 0 \\ \vdots & \vdots& \vdots& \vdots & \vdots & \vdots & \vdots…
user46450
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21
votes
9 answers

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},\\ \sum_{i=1}^n i^2 &= \frac{n(n+1)(2n+1)}{6},\\ \sum_{i=1}^n…
JRowan
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20
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2 answers

What is the best way to solve an equation involving multiple absolute values?

An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases} ax-b & \text{ if } x\ge \frac{b}{a} \\ b-ax & \text{ if } x< \frac{b}{a} \end{cases}$, so an equation with $n$ separate absolute value expressions…
Isaac
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18
votes
1 answer

Getting different answers when integrating using different techniques

Question: Is it possible to get multiple correct results when evaluating an indefinite integral? If I use two different techniques to evaluate an integral, and I get two different answers, have I necessarily done something wrong? Often, an…
Xander Henderson
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17
votes
4 answers

Prove that $A+I$ is invertible if $A$ is nilpotent

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
ugoolm
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16
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2 answers

Finite Sum of Power?

Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^p$$ For $p = 1$, it's just the classic $\frac{n(n+1)}2$. What is it for $p > 1$?
Andrew Tomazos
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16
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3 answers

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 $(a^{-1})^m=e$ for some $m\in \mathbb{Z}$, and we can assume…
ChocolateDroppa
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