# Limit of Series

## Main Question or Discussion Point

Hi,

I don't know how to analyze the following, but I am wondering whether there is a way to determine whether a series of the following form is convergent: V$_{n}$=(V$_{n-1}$+a)/b. Thank you.

jbriggs444
Homework Helper
2019 Award
I don't know how to analyze the following, but I am wondering whether there is a way to determine whether a series of the following form is convergent: V$_{n}$=(V$_{n-1}$+a)/b.
A series can only be convergent (in the usual sense) if the limit of its terms is zero. If a is non-zero, what effect does that have on the limit of the terms in the series?

1 person
pasmith
Homework Helper
Hi,

I don't know how to analyze the following, but I am wondering whether there is a way to determine whether a series of the following form is convergent: V$_{n}$=(V$_{n-1}$+a)/b. Thank you.
This is a linear recurrence relation:

$$V_n - \frac{V_{n-1}}{b} = \frac{a}{b}$$

For $b \neq 1$ the solution is $$V_n = \frac{A}{b^n} + \frac{a}{b-1}$$ for an arbitrary constant $A$. Thus it will converge if and only if $|b| > 1$ or $A = 0$ and its limit will be $\frac{a}{b-1}$.

If $b = 1$ then the solution is $$V_n = A + na$$ for an arbitrary constant $A$, and it does not converge unless $a = 0$.

1 person
Thank you. I see your points. Maybe, in framing this as a limit of a series, I am thinking about the underlying problem incorrectly. So I will expand on the problem I am trying to solve. In a financial context, I am trying to calculate a return that is inclusive of a return-based payment. More specifically, I am trying to calculate a payment (V), which payment is equal to a constant times return, i.e., b*(R/C -1), where R equals final value and C equals initial value. However, final value (R) is defined to include V, such that R= V+a. So the definition is circular. Intuitively, I thought of this as a limit of a series, building inward, where in the first step, the expression would be V = b*(R/C-1) = b*((V+a)/C-1); and in the second step, the expression would be V = b*(((b*(R/C-1))+a)/C-1) = b*(((b*((V+a)/C-1))+a)/C-1). So on and so forth. I would appreciate any guidance as to how to approach this problem. Thanks in advance.

HallsofIvy