166 Theorems for determining the Values of Life Annuities. 
mined accordingly; that is, from a life 2 years older; aud after the 
oroper substitution is made, the result afforded by each particu- 
iH: formula must be combined with the numerical value of the 
expression denoting the expectation of the given life receiving 
Il. m years hence for the value of the annuity in this case. 
Having thus found the value of an annuity deferred for 7 years, 
we may thence derive the value of a similar temporary annuity 
depending on the given life continuing so long in existence. 
I much wish that I could have represented these different 
formulz by others involving the combination of the equal 
single and joint lives. But in each particular instance here ad- 
duced this object could not well be accomplished. 
I shall here introduce the expressions for the sums of a few 
other series that occurred in the course of investigation, and 
which may be found useful on some occasions. 
‘Let X represent 
(o—' 4 52—?7+4+- 92-34 1324-4417 2—54 ....(4n —8)a-*), 
Y (a—' 4507+ 122—3 4 224—4+352—5)..,.(43n?—1])x 5"), 
«And, Z(#—! +17 x—? + 57x—3 + 1 21a — 44+ 209a—5 + oor 
(122?—20n+ 9)c—™. Then will the general expression for the 
sum of each series be respectively, 
4+2—1 —u 
X— rts 9 Chaya 
ac! (x—1) 
(2+327—1) —n Lh US ime —n 
Y= Tae oe) + al It) — (Sn + 2. (Sn2+5n+2))v 
= —_——- ———ooOoOo Or eC —s* 
(z—1)? 
(6+2x-1)—1 —n —1 —n 
Z= pole (ea) + 16.01 + x) +0—1 — (4(6n4 2) +21. (122 + 40+ 1))0 
(@— | 2 z . s 
In regard to the practical mle ate of the above theorems, 
and generally of any other for summing reciprocal series of this 
kind, where the terms of such series are very large, and the rate 
of increase also rapid; it may be observed, that the negative 
powers of (x) should be expanded to a proportionally greater ex- 
tent, in order to obtain a result perfectly accurate. 
The facility and marked attention with which my preceding 
communications were inserted in the Philosophical Magazine, 
have encouraged: me to a further prosecution of these subjects ; 
and I intend at a future opportunity (should I find means—incli- 
nation I possess) to furnish you with a paper embracing the 
discussion of some interesting and rather novel points in the 
doctrine of life assurances, 
Haberdashers-Place. Hoxton, JAS, BENJ, BENWELL, . 
Aug, 15, 1817, 
