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Limit of Series 
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#1
Oct2313, 05:29 PM

P: 43

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[itex]_{n}[/itex]=(V[itex]_{n1}[/itex]+a)/b. Thank you. 


#2
Oct2313, 05:38 PM

P: 963




#3
Oct2313, 05:40 PM

HW Helper
Thanks
P: 1,008

[tex]V_n  \frac{V_{n1}}{b} = \frac{a}{b}[/tex] For [itex]b \neq 1[/itex] the solution is [tex]V_n = \frac{A}{b^n} + \frac{a}{b1}[/tex] for an arbitrary constant [itex]A[/itex]. Thus it will converge if and only if [itex]b > 1[/itex] or [itex]A = 0[/itex] and its limit will be [itex]\frac{a}{b1}[/itex]. If [itex]b = 1[/itex] then the solution is [tex]V_n = A + na[/tex] for an arbitrary constant [itex]A[/itex], and it does not converge unless [itex]a = 0[/itex]. 


#4
Oct2313, 08:21 PM

P: 43

Limit of Series
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 returnbased 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/C1) = b*((V+a)/C1); and in the second step, the expression would be V = b*(((b*(R/C1))+a)/C1) = b*(((b*((V+a)/C1))+a)/C1). So on and so forth. I would appreciate any guidance as to how to approach this problem. Thanks in advance.



#5
Oct2413, 07:47 AM

Math
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PF Gold
P: 39,565

Are you asking about a series (infinite sum) or sequence?



#6
Oct2413, 09:44 AM

P: 43




#7
Oct2513, 07:19 PM

P: 43

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.



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