For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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12
votes
1answer
152 views
+50

if $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$, show that $(\frac{x_{n+1}}{x_{n}}) $ convergent

Let a real positive number sequence $(x_{n})$ such that $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$ and $\sqrt{x_{1}}\geq \sqrt{x_0+1}$. Show that $(\frac{x_{n+1}}{x_{n}}) $ convergent. ...
3
votes
0answers
172 views
+100

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
4
votes
2answers
133 views
+100

How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn[n(\pi^3+1)+\pi^2](n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

We proposed this sum, but we are lacking in knowledge of this area of maths and we would ask if any of the authors would be willing to show us step by step how to go about proving this sum. $$ ...