proofit.

the online resource for solving math

Use power series to find the general solution of the differential equation

A squence of rational numbers is described as follows:

Here the numerator form one sequence, the denominators form a second sequence, and their ratios form a thrid sequence. Let $x_n$ and $y_n$ be, respectively, the numerator and the denimonator of the $n$th fraction $r_n = x_n/y_n$.

1. Verify that $x_1^2-2y_1^2=-1$, $x_2^2-2y_2 = +1$ and more generally that if $a^2-2b^2=-1$ or $+1$, then $(a+2b)^2 - 2(a+b)^2=+1$ or $-1$ respectively.

2. The fractions $r_n = x_n / y_n$ approach a limit as $n$ increases. What is that limit? (Hint: use part (1) to show that $r_n^2-2=\pm (1/y_n)^2$ and that $y_n$ is not less that $n$).

Use series to evaluate the limit