Solve Linear Difference equation

**xinglongdada** · August 6th 2011, 01:33 AM

Dear all, the following ar Berkely problem (Spring 85).

Let

be given. Consider the linear difference equation

(Note the analogy with the differential equation

.)

Find the general solution of by trying suitable exponential substitutions.
Find the solution with and . Denote it by ,
.
Let be fixed and . Show that

**xinglongdada** · August 6th 2011, 01:36 AM

The first one I got the solution:

$y(nh)=(1+h^2)^\frac{n}{2}[C_1 \cos n\phi+C_2 \sin n\phi]$

Here,
$\phi=\arctan h.$

**xinglongdada** · August 6th 2011, 01:39 AM

2. I find that
$S_h(nh)=(1+h^2)^\frac{n-1}{2}\frac{\sin n\phi}{\sin\phi}.$

3. I then confused then. Since I could only verify that
$\lim_{n\to\infty}\sin [n\arctan \frac{x}{n}]=\sin x,$
while
$\lim_{n\to\infty}S_{x/n}(nx/n)=\lim_{n\to\infty}(1+x^2/n^2)^{(n-1)/2}\frac{\sin [n\arctan \frac{x}{n}]}{\sin [\arctan\frac{x}{n}]}=?$

In my calculations, it seems to be \infty. What's going wrong?

Would you help me? Thank you.

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