1

The motiation to this question can be found in About the solution of a difference equation

My question is: Show that any sequence $(u_{n})$ verifying the equation in the above question must tends to infinity as $n→∞$. The expression for $(u_{n})$ is given by:

$$u_{n}=r^{n^2}\left(\sum_{m=1}^{n}\frac{u_1-ru_0+2(m-1)}{r^{m^2}}+u_0\right)$$

Community
  • 1
DER
  • 3,011
  • 1
    The other question assumes that $r\gt4$. // Actually there exists some positive constants $A_r$, $B_r$ and $C_r$ depending only on $r$ such that, when $n\to\infty$, $$u_n-r^{n^2}(A_ru_1+B_ru_0+C_r)\to0.$$ Thus, $u_n\to\infty$ if and only if $A_ru_1+B_ru_0+C_r\ne0$. – Did Jan 18 '15 at 18:21
  • @Did: Can you elaborate with this. In particular, with the existence of those constants epending on $r$. – DER Jan 19 '15 at 06:41
  • 1
    Hint: $$A_r=\sum_{m=1}^\infty r^{-m^2},\qquad B_r=1-rA_r,\qquad C_r=\sum_{m=1}^\infty 2(m-1) r^{-m^2}.$$ – Did Jan 19 '15 at 07:09

0 Answers0