# Tagged Questions

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### Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.

I know that these two are exactly the same equation but I can't seem to prove it. You are also given that $p+q=1$. This is a follow up from a similar question.
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### By plugging $p=1-q$, into the $3$ equations show that $x=y=z$

By plugging $p=1-q$, into the 3 equations: $$\begin{cases} z=py+qx \\ x=pz+qy \\ y=px+qz \end{cases}$$ show that $\boxed{x=y=z}$ This is from the final part of question 7 in this STEP paper, and is ...
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### Series problem:

Parts i) & ii) I can solve. For part iii) I get $z=py+qx$ [For $n=0$] $x=pz+qy$ [For $n=1$] $y=px+qz$ [For $n=2$] leading to $(1-pq)x=(q^2+p)z$ [1] $(1-pq)z=(q^2+p)y$ [2] ...
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### Sequential Algebraic Problem:

The first part i) I can do. For part ii) this is how far I can get: If n is odd then $y=px+qy$ If n is even then $x=py+qx$ After some rearranging i end up with $y(1-q)=px$ & $x(1-q)=py$ and ...
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### Intuition behind a convergent subsequence of $\sin(n)$

$\let\eps\varepsilon$ I was looking through a Real Analysis exam paper one day and was stuck on a question; fortunately there is a solution provided which I will sketch below, but I have no intuition ...
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### Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
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### Finding Similar Sequences

Can we find two sequences: $$\{a (b^0), a (b^1), a (b^2), a (b^3), \dots, a(b^n)\} \bmod p_1$$ $$\{c (d^0), c (d^1), c (d^2), c (d^3), \dots, c(d^n)\} \bmod p_2$$ that differ by only one number? ...
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### Expand $\left(1-\frac2x\right)^{\frac12}$ to find a value of $\sqrt{99}$ and $\sqrt{101}$

Given that $|x|\gt2$ find the first four terms in the series expansion of $\left(1-\frac2x\right)^{\frac12}$ in descending powers of $x$. By taking $x$ = 200 use the series to find a value of ...
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### Concerning $a_{f(n)} = \displaystyle\sum_{i=0}^m{c_i a_{g_i(n)}}$

Suppose $f(n)$ is $2n$. Suppose $g_i(n)$ is $n+i$. In other words, we have a recurrence given by $a_{2n} = c_0 a_n + c_1 a_{n+1} + \dots c_m a_{n+m}$ (1) What is known about this? (2) What type of ...
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### Numerical Methods: Approximating an Integral with Exponentials [closed]

I'd like to know about the best numerical methods for approximating an integral. Unfortunately, I want to know about a fairly general case, so I cannot give a lot of information. Essentially, I have ...
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### Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$

We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$). I was curious to see till what extent I can push the ...
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### Constructing finite versions of arbitrary series

I've been wondering if/when it's possible to "truncate" a series. Example 1 For example, the closed form for the series of naturals is: $\frac{1}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots$ The ...
I'm interested in finding a recursion or simple representation for a "Hadamard product" of two power series. The Hadamard Product The Hadamard product is defined on generating functions $f(x)$ ...
### Is there a relationship between $e$ and the sum of $n$-simplexes volumes?
When I look at the Taylor series for $e^x$ and the volume formula for oriented simplexes, it makes $e^x$ look like it is, at least almost, the sum of simplexes volumes from $n$ to $\infty$. Does ...