INFINITE SERIES I5 



Expressed in words : The end value or sum of a series of pay- 

 ments of $500 each coming every fifteen years continued for ten 

 payments compounded at 3% and the first payment to be made in 

 fifteen years. 



Query : Will the above sum or end value b,e greater or smaller 

 if the first cut comes in five years instead of fifteen years ? 



2. Present value of the sum of a series of periodic payments. 

 The case is the same as under d — 1 and 2. 



Present value is : 



a (i.op" — i) 

 (i.op* — i) I. op"' 



and is modified according to the time when the series begins, or, in 

 the above case, when the first cut is made by writing it : 



a Ciop" — i) lop'-" 

 (i.op'— i) i.op" 



where x is the number of years before the first cut or payment. 



f. Sum of Infinite Series. 



When a farmer buys a farm he really buys the yearly income 

 or rent of the farm for all time, at least there is no set limit. In 

 practice about fifty years' rent is worth as much as the farm, but 

 this fact does not alter the nature of the bargain. 



I. Using the formula as developed under c — 2 and 3, and tak- 

 ing its present value as under d — I, present value: 



a (i.op° — i) 

 (i.op— i) i.op° 



and letting n infinity or 00 the formula becomes : 



a (I.op" — !) 



present value := -; r ^ 



^ (i.op — i) I.op" 



and since i.opoo — i = i.opcc ; present value 



. aCiop") a a 



(i.op — i) I.op" I.op — I ~Top 



If a is the rental per year or $500, and p = 5, then the present 



500 

 value of all the rentals from this farm forever are worth or 



■05 

 $10,000, which simply means that the farm is worth $10,000 if it 

 brings a net income of $500 and the buyer is willing to take 5% on 



