Mathematics for Computer Science

13.1. The Value of an Annuity 507

Proof.

X^1

iD 0

xiWWDnlim!1

Xn

iD 0

xi

D lim

n!1

1 xnC^1

1 x

(by equation 13.2)

D

1

1 x

:

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

In our annuity problem,xD1=.1Cp/ < 1, so Theorem 13.1.1 applies, and we
get

V Dm

X^1

jD 0

xj (by equation 13.3)

Dm

1

1 x

(by Theorem 13.1.1)

Dm


1 Cp

p

.xD1=.1Cp//:

Plugging inmD$50,000 andpD0:08, we see that the valueV is only $675,000.
It seems amazing that a million dollars today is worth much more than $50,000
paid every year for eternity! But on closer inspection, 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 as it turns out, this answer really isn’t so amazing after all.

13.1.5 Examples

Equation 13.2 and Theorem 13.1.1 are incredibly useful in computer science.

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