UpStudy Free Solution:

The approximate value for \(e\), rounded to five decimal places, is:

\[e \approx 2.71828\]

Supplemental Knowledge

The number \(e\) is one of the most important constants in mathematics, often referred to as Euler's number. It is the base of natural logarithms and has numerous applications in calculus, complex analysis, and number theory.

1. Definition of \(e\):

- The constant \(e\) can be defined in several ways. One common definition is through the limit:

\[e = \lim _ { n \to \infty } \left ( 1 + \frac { 1} { n} \right ) ^ n\]

- Another definition involves an infinite series:

\[e = \sum _ { n= 0} ^ { \infty } \frac { 1} { n! } \]

- Here, \(n! \) (n factorial) is the product of all positive integers up to \(n\).

2. Properties of \(e\):

- The function \(e^ x\) (the exponential function) has the unique property that its derivative is equal to itself.

- The natural logarithm, denoted as \(\ln ( x) \), is the inverse function of the exponential function.

3. Applications of \(e\):

- In calculus, \(e^ x\) plays a crucial role in solving differential equations.

- In finance, continuous compound interest calculations use the formula involving \(e^ { rt} \), where \(r\) is the rate and \(t\) is time.

- In probability theory and statistics, distributions such as the normal distribution are expressed using functions involving \(e\).

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