Tagged Questions
2
votes
3answers
50 views
Finding the explicit formula for a recursive sequence, using power series
The Task is to find the explicit expression for the given recursive sequence with the help of power series.
Given:
$a_{0}=0,\ a_{1}=1 \quad$ and $\quad a_{n}=5\cdot a_{n-1} -6\cdot a_{n-2}\quad $ ...
2
votes
2answers
75 views
Using Generating Functions to Solve Recursions
I have the recursion $A(n) = A(n-1) + n^2 - n$ with initial conditions $A(0) = 1$. I attempted to solve it using generating functions and I'm not quite sure I have it right, so I thought I might ask ...
0
votes
3answers
50 views
Power series and recurrence
Please help me find the radius of convergence and the value of the following power series:
$\sum_{n=0}^{\infty} a_nz^n$, when $a_0=1,a_1=-1$, and $3a_n+4a_{n-1}-a_{n-2}=0$ for $n>1$.
2
votes
1answer
98 views
How can I express such function as known functions or power series?
$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$
$$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$
...
3
votes
4answers
138 views
Expansion of $x^4\over1+x^2$ into a power series
I calculated the generating function $A(x)$ of the recurrence $a_n = a_{n-2} - 2a_{n-3}$, $(n \ge 0, a_0 = a_1 = 0, a_2 = 2)$ and I have no clue on how to expand it into a power series in order to ...
13
votes
3answers
396 views
Solving a difficult recursion via generating functions
I have been trying to solve the recurrence:
\begin{align*}
a_{n+1}=\frac{2(n+1)a_n+5((n+1)!)}{3},
\end{align*}
where $a_0=5$, via generating functions with little success. My progress until now is ...
3
votes
3answers
168 views
Recurrence relations problem help
okay im supposed to find a recurrence relation for
$$
a_{n+1}= b \cdot a_n + c \cdot n \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{(1)}
$$
where $b$ and $c$ are constants. the method we learned in class was ...
