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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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4 answers

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

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?
Fan
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3
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2 answers

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.
Shine Mic
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4 answers

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 am supposed to calculate the volume.
user88923
3
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1 answer

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 decided to write one. I was also dissatisfied with…
zwim
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3
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3 answers

Proving $n! > n^3$ for all $n > a$

Prove by induction: Find a, and prove the postulate by mathematical induction. $$\text{For all}~ n > a,~ n! > n^3$$ Where ! refers to factorial. So far I've done a bit of it, I'll skip right to the inductive statement and assume that $k! > k^3$,…
Xian
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3
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7 answers

Alternate ways to prove that $4$ divides $5^n-1$

I was working for various method to solve this: For all $n\in \mathbb N$: $4\;\mid\;(5^{n}-1)$. My try was: 1st: $$n=1 \to 4|5^1-1\\n \geq 2 \to 5^n=25,125,625,3125,...\\ n\geq 2 \to…
Khosrotash
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5 answers

Find a closed form for $\sum_{k=0}^{n} k^3$

Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.
Boris Ablamunits
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2 answers

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 $(x,x) \not\in R$ for all $x \in X$, transitive if $(x,y) \in R$ and…
Xander Henderson
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2
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5 answers

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 modulo $n$ there are $n$ whose sum is the…
John Omielan
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2
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0 answers

Proof of reduced upper-tail inequality for standard normal distribution

X∼N(0,1), then to prove that for x>0, $$ P(X>x)≤ \frac{1}{2}exp(−x^2/2) $$ I know how to prove the other two kinds of upper-tail inequality for standard normal distribution like this one $$exp(−x^2/2) $$ and this one $$…
Juen Guo
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4 answers

Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$ (Negative Binomial Theorem)

I was wondering if there is also a closed expression for the series $$\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$$ where $|x|<1.$ A few examples suggest that the answer is $\frac{1}{(1-x)^{k+1}}$ but I don't see how to show this.
user296837
1
vote
1 answer

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.
avz2611
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1
vote
1 answer

Limit of an arithmetic average series

Sorry in advance as English is not my primary language. I randomly thought of the following simple problem, and I coudn't solve it after one one hour trying. Maybe you guys can help. Let $a_1$ and $a_2$ be positive real numbers. Let $a_n$ be the…
Anderson Linhares
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vote
2 answers

Find $11^{644} \mod 645$

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 with the solution is attached. Thanks in…
Yusha
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