Answer:
[tex]lim_{n \to \infty} A_n = \frac{R}{i}[/tex]
Step-by-step explanation:
For this case we have this expression:
[tex] A_n = R [\frac{1 -(1+i)^{-n}}{i}][/tex]
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:
[tex] lim_{n \to \infty} A_n[/tex]
So we have this:
[tex] lim_{n \to \infty} A_n = lim_{n \to \infty}R [\frac{1 -(1+i)^{-n}}{i}] [/tex]
Then we can do this:
[tex] lim_{n \to \infty} A_n = lim_{n \to \infty} R [\frac{1 -\frac{1}{(1+i)^n}}{i}][/tex]
[tex]lim_{n \to \infty} A_n = R lim_{n \to \infty} [\frac{1 -\frac{1}{(1+i)^n}}{i}][/tex]
And after find the limit we got:
[tex] lim_{n \to \infty} A_n = R [\frac{1-0}{i}][/tex]
Becuase : [tex] \frac{1}{(1+i)^{\infty}} =0[/tex]
And then finally we have this:
[tex]lim_{n \to \infty} A_n = \frac{R}{i}[/tex]
