# A recreational series problem

• elgen
In summary, the conversation discusses the problem of finding the input impedance of a series of cascaded resistors and how to approach it. The main question is how to get rid of the product and summation terms in order to find an expression for Zn+1, R1, R2, and n. The suggestion is to use the limit as Zn+1 approaches Zn to find the final expression.

#### 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

hi elgen!

the question only asks for the limit

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

That sounds cool. Thx for the pointer!

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