Find Out How to Analyze V_n Series

  • Thread starter Thread starter elementbrdr
  • Start date Start date
  • Tags Tags
    Series
AI Thread Summary
The discussion centers on analyzing the convergence of a sequence defined by the recurrence relation V_n = (V_{n-1} + a) / b. It is established that the sequence converges if |b| > 1 or if the arbitrary constant A is zero, with the limit being a/(b-1). If b = 1, convergence only occurs when a = 0. The original poster clarifies that they are interested in the sequence's behavior after infinite iterations, not the sum of its terms, and seeks a simpler expression for the calculation involved in their financial context. The conversation highlights the importance of distinguishing between sequences and series in mathematical analysis.
elementbrdr
Messages
43
Reaction score
0
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.
 
Mathematics news on Phys.org
elementbrdr said:
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?
 
  • Like
Likes 1 person
elementbrdr said:
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.
 
  • Like
Likes 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.
 
Are you asking about a series (infinite sum) or sequence?
 
  • Like
Likes 1 person
HallsofIvy said:
Are you asking about a series (infinite sum) or sequence?

I think that what I am trying to do falls within the category of a sequence, rather than a series. I am interested in what the value of the expression would be after an infinite number of iterations of the "steps" illustrated above (and not in the sum of the expressions produced by each step). Sorry to have started off the thread with a misconception.
 
Viewing my question in terms of a sequence rather than a series, is there a way to state the calculation I described as a simple expression? I would be happy to try to clarify further if my question is still muddled. Thank you.
 
Back
Top