3
votes
find the expansion of $\frac{1}{1-(\frac{nx}{1+x})}$
You can write $$\dfrac1{1-\frac{nx}{x+1}}=\frac{1+x}{1-(n-1)x}=(1+x)\sum_{k=0}^\infty (n-1)^kx^k$$
Then combine like terms to get $$1+\sum_{k=1}\left((n-1)^k+(n-1)^{k-1}\right)x^k$$
And $(n-1)^k+(n-1)^...
- 180k
3
votes
find the expansion of $\frac{1}{1-(\frac{nx}{1+x})}$
\begin{eqnarray*}
\frac{1}{1-\frac{1+nx}{1+x}} &=& \frac{1+x}{1-(n-1)x} \\
&=& 1+ \frac{nx}{1-(n-1)x} \\
\end{eqnarray*}
Now Geometrically expand.
3
votes
Accepted
Is $\frac{1}{1-x}$ the "best" we can do to create a sequence of polynomials that share both coefficients and roots?
Write the divisibility conditions as $$P_{n+1}=fP_n\qquad\text{and}\qquad P_{n+1}-P_n=gx^{d_n+1}$$ for some nonzero $f,g\in\mathbb Z[x]$ with $\deg(f)>0$. Substituting $P_{n+1}$, this yields \begin{...
- 2,703
2
votes
Accepted
Power series for rational function
It seems there is a small mistake when doing the partial fraction decomposition. We have
\begin{align*}
\frac{1}{1-3x-3x^2}&=\left(-\frac{1}{3}\right)\frac{1}{x^2+x-\frac{1}{3}}\\
&=-\frac{1}{...
- 111k
1
vote
Irreducible and composite formal power series
Here are some series that are irreducible (and hence not composite) in your sense:
Call a series small if it is a polynomial, and large if it is not. Then you can see that a product of anything non-...
- 9,250
1
vote
If $n\in\Bbb N$ is factor of 144, and $xy=n$ , then no. of ordered pairs of $(x,y)$?- ($x,y\in\Bbb N$)
Any factor $n$ of $144$ is of the form $2^a\cdot 3^b$, where $0 \leq a \leq 4$ and $0 \leq b \leq 2$. If $xy=n$, then $x=2^{x_1}\cdot3^{x_2}$ and $y=2^{y_1} \cdot 3^{y_2}$. Thus,
$$xy=2^{x_1+y_1}\...
- 42k
1
vote
Accepted
What’s the coefficient of $n^{k-i}$ in $\Delta^jn^k$?
Firstly, in the quoted statement from Wilf, Zeilberger & Petkovšek's book the term $O(n^{k-j-1})$ is an example of asymptotic notation, which in this context means there are some terms left out of ...
- 514
1
vote
Find the number of all subsets of $\{1,\dots,N\}$ with $n$ elements such that the sum of the elements is divisible by $N$.
We have
$$ \large
f(\omega^j, x) = \prod_{k=1}^N (1+x\omega^{jk}) \\
= (-x)^N \prod_{k=1}^N ((-x)^{-1}-\omega^{jk}) \\
= (-x)^N \left(\prod_{k=1}^{ \frac{N}{\gcd(N,j)} } ((-x)^{-1}-\omega^{jk})\right)^...
- 10.9k
1
vote
solve a recursion formula using a generating function
For $Z\in M_2(\mathbb{R})$ satisfying $Z^2=AZ+B$ we have $$Z^n=AZ^{n-1}+BZ^{n-2},\quad n\ge 2$$ For $$Z^n=\begin{pmatrix} a_n &*\\ b_n&*\end{pmatrix}$$ the sequence $v_n= \, ^t(a_n,b_n)$ ...
- 39.5k
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