Question

If an annuity makes an infinite series of equal payments at the end of the interest periods, it is called a perpetuity. If a lump sum investment of An is needed to result in n periodic payments of R when the interest rate per period is i, then the sum is represented by the following.

An = R 1 - (1 + i)^-n
i


Evaluate the following limit to find a formula for the lump sum payment for a perpetuity.

lim An
n?

Answer 1

`lim_(n \to \infty) A_n = (R)/(i)`

Explanation:

For this case we have this expression:

`A_n = R [(1 -(1+i)^(-n))/(i)]`

The lump sum investment of An is needed to result in n periodic payments of R when the interest rate per period is i.

And we want to find the:

`lim_(n \to \infty) A_n`

So we have this:

`lim_(n \to \infty) A_n = lim_(n \to \infty)R [(1 -(1+i)^(-n))/(i)]`

Then we can do this:

`lim_(n \to \infty) A_n = lim_(n \to \infty) R [(1 -(1)/((1+i)^n))/(i)]`

`lim_(n \to \infty) A_n = R lim_(n \to \infty) [(1 -(1)/((1+i)^n))/(i)]`

And after find the limit we got:

`lim_(n \to \infty) A_n = R [(1-0)/(i)]`

Becuase :
`(1)/((1+i)^(\infty)) =0`

And then finally we have this:

`lim_(n \to \infty) A_n = (R)/(i)`