Definition and Rules for Differentiating Power Series in Calculus

📚 Definition of a Power Series

A power series is an infinite series of the form:

$\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3 + \cdots$

Where:

• 🔢 $x$ is a variable.
• 📍 $a$ is a constant, representing the center of the series.
• 📈 $c_n$ are constants, representing the coefficients of the series.

Power series are fundamental in calculus and analysis because they provide a way to represent functions as infinite polynomials. They are used extensively in solving differential equations, approximating functions, and evaluating integrals.

📜 History and Background

The concept of power series dates back to the 17th century with the work of mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz. Newton used power series to approximate solutions to equations and to develop his theory of calculus. Brook Taylor formally introduced Taylor series, a specific type of power series, in 1715. Colin Maclaurin later popularized a special case of Taylor series centered at zero, known as the Maclaurin series.

• 🕰️ Early Development: Isaac Newton and Gottfried Wilhelm Leibniz laid the groundwork.
• ✍️ Formalization: Brook Taylor introduced Taylor series in 1715.
• 🌟 Special Case: Colin Maclaurin popularized Maclaurin series.

🔑 Key Principles for Differentiating Power Series

Differentiating power series involves applying the standard rules of differentiation term by term. Given a power series:

$\sum_{n=0}^{\infty} c_n (x-a)^n$

Its derivative is:

$\frac{d}{dx} \sum_{n=0}^{\infty} c_n (x-a)^n = \sum_{n=1}^{\infty} n \cdot c_n (x-a)^{n-1}$

Here are the key principles:

• ➕ Term-by-Term Differentiation: Differentiate each term in the series individually.
• ✍️ Power Rule: Apply the power rule of differentiation: $\frac{d}{dx} x^n = n \cdot x^{n-1}$.
• 🔄 Index Adjustment: Adjust the index of summation if necessary to simplify the resulting series.
• 🛑 Constant Term: The derivative of the constant term $c_0$ is zero.
• 📍 Center of Series: The center of the series, $a$, remains unchanged during differentiation.
• 💡 Radius of Convergence: The radius of convergence remains the same after differentiation, although the interval of convergence may change.

⚙️ Steps for Differentiating Power Series:

1. Write down the power series: $\sum_{n=0}^{\infty} c_n (x-a)^n$.
2. Apply the power rule to each term: $\frac{d}{dx}[c_n (x-a)^n] = n \cdot c_n (x-a)^{n-1}$.
3. Write out the differentiated power series: $\sum_{n=1}^{\infty} n \cdot c_n (x-a)^{n-1}$. Note that the summation starts from $n=1$ since the derivative of the constant term ($n=0$) is zero.
4. If necessary, re-index the series to match a desired form or to simplify further calculations. For instance, you might want to shift the index $n$ to $n+1$ to get a series in terms of $(x-a)^n$.

🧮 Real-World Examples

Example 1: Differentiating a Simple Power Series

Consider the power series:

$\sum_{n=0}^{\infty} x^n$

This is a geometric series that converges to $\frac{1}{1-x}$ for $|x| < 1$. Let's differentiate it:

$\frac{d}{dx} \sum_{n=0}^{\infty} x^n = \sum_{n=1}^{\infty} n \cdot x^{n-1} = 1 + 2x + 3x^2 + 4x^3 + \cdots$

This resulting series converges to $\frac{1}{(1-x)^2}$ for $|x| < 1$, which is the derivative of $\frac{1}{1-x}$.

• ➕ Original Series: $\sum_{n=0}^{\infty} x^n$
• ➗ Differentiated Series: $\sum_{n=1}^{\infty} n \cdot x^{n-1}$
• 🎯 Result: Derivative of $\frac{1}{1-x}$ is $\frac{1}{(1-x)^2}$

Example 2: Differentiating a Taylor Series

Consider the Taylor series for $e^x$:

$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

Differentiating term by term:

$\frac{d}{dx} e^x = \frac{d}{dx} \sum_{n=0}^{\infty} \frac{x^n}{n!} = \sum_{n=1}^{\infty} \frac{n \cdot x^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!}$

By re-indexing (let $k = n-1$), we get:

$\sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$

Thus, the derivative of $e^x$ is $e^x$, as expected.

• 🧬 Taylor Series: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
• 🧪 Differentiated Series: $\sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!}$
• 🎯 Result: Derivative of $e^x$ is $e^x$

📝 Practice Quiz

1. Differentiate the power series: $\sum_{n=0}^{\infty} (n+1)x^n$
2. Differentiate the power series: $\sum_{n=1}^{\infty} \frac{x^n}{n}$
3. Differentiate the power series: $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$

🔑 Solutions

1. $\sum_{n=1}^{\infty} n(n+1)x^{n-1}$
2. $\sum_{n=1}^{\infty} x^{n-1}$
3. $\sum_{n=1}^{\infty} \frac{(-1)^n 2n x^{2n-1}}{(2n)!}$ = $\sum_{n=1}^{\infty} \frac{(-1)^n x^{2n-1}}{(2n-1)!}$

✅ Conclusion

Differentiating power series is a powerful technique in calculus that allows us to find derivatives of functions represented as infinite sums. By applying the basic rules of differentiation term by term, we can obtain new power series that represent the derivatives of the original functions. Understanding the key principles and practicing with examples will help you master this important concept. Remember to pay attention to the index of summation and the interval of convergence to ensure accurate results. Happy differentiating! 🚀