Which conditions should I impose to make the sequence of partial sums be convergent?

Question

I have the following equation $x_{n+2} + (a − b)x_{n+1} + (a + b)x_n = 0$, which conditions should I impose to a and b to make the $S_n$ sequence of partial sums of a solution convergent ?($S_n=x_0+x_1+...+x_n$)

Answer 1

The equation is linear: if $\{x_i\}, \{y_i\}$ are two sequences that obey it, and $A, B$ are constants, then $\{Ax_i + By_i\}$ also obeys it. Thus the set of solutions is a vector space. And because each solution is completely determined by the values of $x_0, x_1$, the vector space is 2-dimensional. Thus if you can find two independent solutions, all other solutions are linear combinations of those two.

For $x_n = r^n$ to be a solution, it must be that $r^2 + (a-b)r + (a+b) = 0$ or $$r = \frac {b-a \pm \sqrt{(a-b)^2 - 4(a+b)}}2$$ These two values of $r$ provide two independent solutions. Calling the values $r_1, r_2$, every solution is of the form $x_i = Ar_1^i + Br_2^i$, so the $n$-th partial sum is $$S_n = A\sum_{i=0}^n r_1^i + B\sum_{i=0}^n r_2^i$$ from which it should be easy to see what is needed to ensure all such sums converge.

Warning: There is a hole in the argument above. Spot it, and we can discuss how it can be closed.