What is the Limit of Z_n as n Approaches Infinity in Cascaded Resistors?

[elgen]

Dear all,

The problem of finding the input impedance of a series of cascaded resisters can be stated as

$$Z_{n+1}=R_1+\frac{R_2 Z_n}{R_2+Z_n}$$ where $$Z_1=R_1+R_2$$. What is $$\lim_{n\to\infty}Z_n$$?

My attempt is to re-write the recurrance relation as
$$(R_2+Z_n)Z_{n+1}-(R_1+R_2)Z_n-R_1R_2=0$$

which is
$$R_2 Z_{n+1} + Z_{n}Z_{n+1} - (R_1+R_2)Z_n - R_1 R_2 = 0$$

$$R_2 Z_{n} + Z_{n-1}Z_{n} - (R_1+R_2)Z_{n-1} - R_1 R_2 = 0$$
...
$$R_2 Z_2 + Z_1 Z_2 - (R_1+R_2)Z_1 - R_1 R_2 = 0$$

Summing them up gives
$$R_2(Z_{n+1}-Z_1)+Z_{n+1}Z_n + ... + Z_2 Z_1 - R_1(Z_n +...+Z_1) - nR_1R_2=0$$.

I am not sure on how to get rid of the product terms and summation terms to get an expression of only $$Z_{n+1}, R_1, R_2$$ and $$n$$. Any suggestion on possible attack?

Thank you.

elgen

 

[tiny-tim]

hi elgen!

the question only asks for the limit …

you can find the limit (if it exists) by putting Z_n+1 = Z_n

 

[elgen]

That sounds cool. Thx for the pointer!