# Differentiation of Infinite series

### Differentiation of Infinite Series:

• Infinite series refers to the sum of an infinite sequence of terms. Differentiating such series involves differentiating term by term.
• nfinite series represents an infinite sum of terms involving a variable $x$, often written in the form of $f\left(x\right)={\sum }_{n=0}^{\mathrm{\infty }}{a}_{n}{x}^{n}$, where ${a}_{n}$ are coefficients and $x$ is the variable.

### Conditions for Differentiation of Infinite Series:

1. Uniform Convergence:

• To differentiate an infinite series term by term, the series must converge uniformly within a given interval.
• Uniform convergence ensures that the derivative of the sum is equal to the sum of the derivatives of each term.
2. Continuity and Differentiability:

• Each term of the series should be continuous and differentiable within the interval of interest for the differentiation to hold.

### Differentiation of Power Series:

• Power series is a type of infinite series that can be differentiated and integrated term by term within its interval of convergence.

### Steps for Differentiating Infinite Series:

1. Write the function as a power series:

• Express the function as an infinite sum involving powers of $x$.
2. Differentiate each term individually:

• Differentiate each term of the series with respect to $x$ using standard differentiation rules.
• Derivative of ${x}^{n}$ is $n{x}^{n-1}$.
3. Find the resulting series:

• Sum up the individual derivatives to get the derivative of the original series.

### Example:

Consider the series $f\left(x\right)={\sum }_{n=1}^{\mathrm{\infty }}\frac{{x}^{n}}{n}$ within the interval $\left(-1,1\right)$.

1. Express the series as a function:

• $f\left(x\right)={\sum }_{n=1}^{\mathrm{\infty }}\frac{{x}^{n}}{n}=-\mathrm{ln}\left(1-x\right)$, $-1
2. Differentiate term by term:

• Differentiate $f\left(x\right)$ term by term to find its derivative: ${f}^{\mathrm{\prime }}\left(x\right)={\sum }_{n=1}^{\mathrm{\infty }}{x}^{n-1}=\frac{1}{1-x}$, $-1
3. Verify the convergence and interval:

• The derivative ${f}^{\mathrm{\prime }}\left(x\right)$ converges within the interval $\left(-1,1\right)$ and is equal to the sum of the derivatives of each term in the original series.

### Example:

Given the function $f\left(x\right)={\sum }_{n=0}^{\mathrm{\infty }}\frac{{x}^{n}}{n!}$ which represents ${e}^{x}$:

1. Express the function as a power series:

• $f\left(x\right)={e}^{x}$ is a well-known power series.
2. Differentiate each term individually:

• Differentiate each term of the power series ${e}^{x}$ term by term.
• $\frac{d}{dx}\left(\frac{{x}^{n}}{n!}\right)=\frac{n{x}^{n-1}}{n!}=\frac{{x}^{n-1}}{\left(n-1\right)!}$
3. Find the resulting series:

• The derivative of $f\left(x\right)$ is $\frac{d}{dx}{e}^{x}={\sum }_{n=0}^{\mathrm{\infty }}\frac{{x}^{n-1}}{\left(n-1\right)!}$.

### Tips for Differentiating Infinite Series:

• Term-by-Term Differentiation:

• Differentiate each term of the series individually as if it were a separate function.
• Convergence Analysis:

• Ensure that the resulting series after differentiation converges within the same interval as the original series.
• Uniform Convergence:

• Confirm uniform convergence to ensure that differentiation can be performed term by term.

### Special Cases:

• Power Series:

• Often used in differentiating functions represented as power series expansions, such as Taylor or Maclaurin series.
• Functions with Special Forms:

• Differentiating trigonometric, exponential, or logarithmic series often involves specific differentiation rules tailored to these functions.

### Advantages of Differentiating Infinite Series:

• Extension of Differentiation Rules:

• Allows differentiation of functions represented as infinite series using standard differentiation rules.
• Representation of Functions:

• Infinite series often represent functions and their derivatives, aiding in function manipulation and analysis.