Proving the Derivative and Expansion of (e^x) Through Binomial Theorem and Power Series

In this article, we explore the mathematical foundations of the exponential function ex. Specifically, we will delve into the proof of its derivative being ex itself and its power series expansion using the binomial theorem. We will utilize diagrams and detailed explanations to make the concepts more accessible.

1. Binomial Theorem and Its Application to (e^x)

The binomial theorem is a powerful tool in algebra and is defined for any real number n and any variable x as follows:

(1 xn)n∑k0nn^{UNDERLINE{}kx^kk!InvisibleTimes;n^kleft(1 frac{x}{n}right)^n sum_{k0}^{n} frac{n^{underline{k}} x^k}{k! n^k}

Here, the underlined exponent (n^{underline{k}}) denotes the falling factorial, which is defined as (n(n-1)(n-2)cdots(n-k 1)).

2. Deriving (e^x) from the Binomial Expansion

Now, let's consider the limit as (n) approaches infinity:

exlimn→∞(1 xn)nlimn→∞∑∑k0nn^{UNDERLINE{}kx^kk!InvisibleTimes;n^k∑∑k0∞xkk!e^x lim_{n to infty} left(1 frac{x}{n}right)^n lim_{n to infty} sum_{k0}^{n} frac{n^{underline{k}} x^k}{k! n^k} sum_{k0}^{infty} frac{x^k}{k!}

This result shows that the exponential function (e^x) can be represented as an infinite power series, which is derived from the binomial expansion.

3. Graphical Representation and Accuracy

To further understand the accuracy of the power series expansion, we can look at a pictorial representation. The graph below shows the power series expansion of (e^x) with varying numbers of terms, compared to the exact function (e^x).

The blue curve represents the exact function (e^x). The red curve shows the partial sum of the power series. As more terms are added, the red curve converges more closely to the blue curve, illustrating the accuracy and convergence of the power series.

4. Historical Perspective and Definition of (e)

The number (e) is a fundamental constant in mathematics. It represents continuous compounding interest. Let's derive (e) from this principle:

The basic formula for compound interest is:

AP(1 rn)^ntA Pleft(1 frac{r}{n}right)^{nt}

Where:

A final amount after undergoing compound interest P principal investment amount (initial deposit or loan amount) r annual interest rate (in decimal form) n number of times that interest is compounded per year t number of years the money is invested or borrowed for

For simplicity, let P 1, r 1, and t x. The formula then becomes:

A(1 1n)^nxA (1 frac{1}{n})^{nx}

As n approaches infinity, (1 frac{1}{n}) becomes e. Therefore:

AexA e^x

5. Conclusion

Thus, we've shown how the binomial theorem leads to the power series expansion of (e^x), and why this expansion is so crucial in understanding and applying the exponential function in various fields of mathematics and science. The accurate representation of (e^x) through a series of terms is a testament to the elegance of mathematical concepts and their real-world applications.