Sequences — Question 4

*Question 4 Let $a_n=\left(1+\dfrac2n\right)^n$.

1. Interpret $a_n$ as the balance produced by compounding a principal of $1$ at a total rate of $2$ over one time unit, using $n$ periods.

2. Evaluate $\lim_{n\to\infty}a_n$ using the standard exponential limit.

3. Give a decimal approximation of the limiting balance.

*Question 4 – Solution

The principal is $P=1$. Dividing the total rate $2$ into $n$ equal periods gives rate $2/n$ per period; applying that growth factor $n$ times produces $1(1+2/n)^n$.

Put $m=n/2$. Then $m\to\infty$ and $$a_n=\left[\left(1+\frac1m\right)^m\right]^2.$$

Using $\lim_{m\to\infty}(1+1/m)^m=e$ and continuity of the squaring function gives $$\lim_{n\to\infty}a_n=e^2\approx7.38906.$$

This agrees with the continuous-compounding formula $Pe^{rt}=e^2$ for $P=1$, $r=2$, and $t=1$. The standard exponential limit supplies the proof.