Mathematics for Computer Science

14.1. The Value of an Annuity 405

Proof.

X^1

iD 0

xiWWDnlim!1

Xn

iD 0

xi

D lim

n!1

1 xnC^1

1 x

(by equation 14.2)

D

1

1 x

:

The final line follows from that fact that limn!1xnC^1 D 0 whenjxj< 1. 

In our annuity problem,xD1=.1Cp/ < 1, so Theorem 14.1.1 applies, and we
get

V Dm

X^1

jD 0

xj (by equation 14.3)

Dm

1

1 x

(by Theorem 14.1.1)

Dm


1 Cp

p

.xD1=.1Cp//:

Plugging inmD$50,000 andpD0:08, we see that the valueV is only $675,000.
Amazingly, a million dollars today is worth much more than $50,000 paid every
year forever! Then again, if we had a million dollars today in the bank earning 8%
interest, we could take out and spend $80,000 a year forever. So on second thought,
this answer really isn’t so amazing.

14.1.5 Examples

Equation 14.2 and Theorem 14.1.1 are incredibly useful in computer science.

Here are some other common sums that can be put into closed form using equa-