Compounding with a 5% Interest Rate
A man invests \$1000 in an account with a 5% annual interest rate. He knows that money in an account where interest is compounded semiannually will earn interest faster than money in an account where interest is compounded annually. He wonders how much interest can be earned by compounding it more and more often. In this problem we investigate his question.
If the man's interest is compounded annually, his yearend balance will be: $$\begin{align}\$1000 + 5 \% \cdot \$1000 &= \$1000 + 0.05 \cdot \$1000 \\ &= \$1000(1+0.05)\\ &= \$1050.\end{align}$$
If his interest is compounded semiannually, he earns half the annual interest at midyear, and his midyear balance is: $$\begin{align} \$1000 + \frac{5 \%}{2} \cdot \$1000 &= \$1000 + \frac{0.05}{2} \cdot \$1000\\ &= \$1000\left(1+\frac{0.05}{2}\right)\\ &= \$1025.\end{align}$$
At yearend he earns the other half of his annual interest and his yearend balance is: $$\begin{align}\$1025+ \frac{5 \%}{2} \cdot \$1025 &= \$1025 + \frac{0.05}{2} \cdot \$1025\\ &= \$1025\left(1+\frac{0.05}{2}\right)\\&= \$1000\left(1+\frac{0.05}{2}\right)\left(1 +\frac{0.05}{2}\right)\\& = \$1000\left(1 +\frac{0.05}{2}\right)^2 \\& =\$1050.625.\end{align}$$
 Find the end of year balance if the interested is compounded quarterly.
 Write an expression which gives the man's end of year balance in terms of the number of times the interest is compounded, $n$.
Substitute $k=\frac{0.05}{n}$ into your expression so that the whole expression is written in terms of $k$ instead of in terms of $n$.'
Now we'll investigate what happens to the end of year balance as we compound the interest more and more. This means that we want to increase the value of $n$. What does increasing the value of $n$ do to the value of $k$?

Complete the table below to help you see what happens to the end of year balance as $k$ becomes larger and larger. Round to the 5th decimal place.
$k$ $(1+k)^{\frac1k}$ 0.1 0.01 0.001 0.0001 0.00001 The values in the second column of your table should not appear to be growing out of control. They should appear to approach a limiting value. This value is an irrational number which mathematicians denote with the letter $e$.
Based on the results of your table, what value does it appear the end of year balance will approach as the interest is compounded more and more often? Write this value in terms of $e$.