Intereting Posts

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I have learned that 1/0 is infinity, why isn't it minus infinity?
How many sequences of $n$ tosses of a coin that do not contain two consecutive heads have tails as the first toss?
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Determining Laurent series $f(z)=\frac{1}{(z-2)(z-3)}$.
$A^TA$ is always a symmetric matrix?
Complete induction proof that every $n > 1$ can be written as a product of primes
History of Commutative Algebra
Splitting a renewal process
$k$-element subsets of $$ that do not contain $2$ consecutive integers
Can any two disjoint nonempty convex sets in a vector space be separated by a hyperplane?
One last question on the concept of limits
Prove that $a^{2^n}=1 \mod 2^{n+2}$

Given an infinite sequence of numbers, *first differences* denote a sequence of numbers that are pairwise differences, *second differences* denote a new sequence of pairwise differences of this sequence, and so on.

```
1 2 4 7 11 16 22 29
1 2 3 4 5 6 7 -- first differences
1 1 1 1 1 1 -- second differences
0 0 0 0 0 -- third differences
```

Sequences of *n*^{th} powers of numbers exhibit an interesting property. Take a sequence of squares: (1,4,9,16,25,…).

- First differences: (3,5,7,9,11…)
- Second differences: (2,2,2,2,2…)

Take a sequence of cubes: (1,8,27,64,125…)

- Is $\sum\limits_{n=1}^{\infty}\frac{n!}{n^n}$ convergent?
- Given number of trailing zeros in n!, find out the possible values of n.
- Evaluate a finite sum with four factorials
- Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$
- How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?
- Showing $\binom{2n}{n} = (-4)^n \binom{-1/2}{n}$

- First differences: (7,19,37,61,91…)
- Second differences: (12,18,24,30,36…)
- Third differences: (6,6,6,6,6…)

Take a sequence of powers of 4: (1,16,81,256,625,…)

- First differences: (15,65,175,369,671,…)
- Second differences: (50,110,194,302,434,…)
- Third differences: (60,84,108,132,156,…)
- Fourth differences: (24,24,24,24,24,…)

1, 2, 6, 24… that’s clearly a factorial! So a sequence of powers of *n* will have a sequence of repeating *n!* as its *n*^{th} differences. How do I prove this relation holds? Bonus points if you give just enough theoretical information for a non-mathematician, but leave the proof as an exercise.

Note:It actually doesn’t matter if our sequence includes negative numbers too, sequence (…,-2,-1,0,1,2,…) still has (…,1,1,1,1,1,…) as its first differences.

- How is Ramanujan's recurrence relation for his nested radical solved?
- Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
- Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$
- Number of sequences with n digits, even number of 1's
- How to solve this recurrence Relation - Varying Coefficient
- Finding all the numbers that fit $x! + y! = z!$
- Maxima and minima of $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}$.
- How do I resolve a recurrence relation when the characteristic equation has fewer roots than terms?
- Factorial of a large number and Stirling approximation
- non homogeneous recurrence relation

The difference sequence of any polynomial $Cx^n + \cdots$ (where here and elsewhere, $\cdots$ hides terms of lower degree) is of the form $Cnx^{n-1} + \cdots$, since $(x+1)^n – x^n = nx^{n-1} + \cdots$ (just open the binomial). If we start with $x^n + \cdots$ then we get $nx^{n-1} + \cdots$, then $n(n-1)x^{n-2} + \cdots$, then $n(n-1)(n-2)x^{n-3} + \cdots$, and eventually, after $n$ times, $n(n-1)(n-2)\cdots 1 x^0 = n!$. You can prove this by induction.

- Partition of N into infinite number of infinite disjoint sets?
- Proving that $\frac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\frac{(-1)^{(p_{k}-1)/2}}{p_{k}} \right )^{-1}$ an identity of Euler's.
- A Hilbert basis for $L^2 (\times)$
- Seeking for a hint to a limit question $\lim_{n \to \infty}(a_{n+1}^{\alpha}-a_n^{\alpha})=0$
- $\Delta^ny = n!$ , difference operator question.
- How to perform a fair coin toss experiment over phone?
- How to prove those “curious identities”?
- How do collinear points on a matrix affect its rank?
- Example of a first-countable non-$T_1$ compact topological space which is not sequentially compact?
- Decomposable elements of $\Lambda^k(V)$
- If every compact set is closed, then is the space Hausdorff?
- Show $x^6 + 1.5x^5 + 3x – 4.5$ is irreducible in $\mathbb Q$.
- Proving a relation between $\sum\frac{1}{(2n-1)^2}$ and $\sum \frac{1}{n^2}$
- Symplectic group action
- If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$