Question

select the answers that best complete the given statement.
the value that ((1+\frac{1}{n})^n) approaches as (n) gets larger and larger is the irrational number (e), called the (square) base. this irrational number is approximately equal to (square) (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Recall the definition of the number \(e\)

The number \(e\) is defined as \(\lim_{n
ightarrow\infty}(1 + \frac{1}{n})^{n}\).

Step2: Calculate the approximate value

When \(n = 1000\), \((1+\frac{1}{1000})^{1000}\approx2.71692\). When \(n = 10000\), \((1+\frac{1}{10000})^{10000}\approx2.71815\). As \(n\) gets larger, \((1 + \frac{1}{n})^{n}\) approaches \(e\approx2.72\) (rounded to two decimal places).

Answer:

\(2.72\)