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 $f(x)={\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}{x}^{n}$, where ${a}_{n}$ are coefficients and $x$ is the variable.
Conditions for Differentiation of Infinite Series:

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.

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:

Write the function as a power series:
 Express the function as an infinite sum involving powers of $x$.

Differentiate each term individually:
 Differentiate each term of the series with respect to $x$ using standard differentiation rules.
 Derivative of ${x}^{n}$ is $n{x}^{n1}$.

Find the resulting series:
 Sum up the individual derivatives to get the derivative of the original series.
Example:
Consider the series $f(x)={\sum}_{n=1}^{\mathrm{\infty}}\frac{{x}^{n}}{n}$ within the interval $(1,1)$.

Express the series as a function:
 $f(x)={\sum}_{n=1}^{\mathrm{\infty}}\frac{{x}^{n}}{n}=\mathrm{ln}(1x)$, $1<x\le 1$

Differentiate term by term:
 Differentiate $f(x)$ term by term to find its derivative: ${f}^{\mathrm{\prime}}(x)={\sum}_{n=1}^{\mathrm{\infty}}{x}^{n1}=\frac{1}{1x}$, $1<x<1$

Verify the convergence and interval:
 The derivative ${f}^{\mathrm{\prime}}(x)$ converges within the interval $(1,1)$ and is equal to the sum of the derivatives of each term in the original series.
Example:
Given the function $f(x)={\sum}_{n=0}^{\mathrm{\infty}}\frac{{x}^{n}}{n!}$ which represents ${e}^{x}$:

Express the function as a power series:
 $f(x)={e}^{x}$ is a wellknown power series.

Differentiate each term individually:
 Differentiate each term of the power series ${e}^{x}$ term by term.
 $\frac{d}{dx}\left(\frac{{x}^{n}}{n!}\right)=\frac{n{x}^{n1}}{n!}=\frac{{x}^{n1}}{(n1)!}$

Find the resulting series:
 The derivative of $f(x)$ is $\frac{d}{dx}{e}^{x}={\sum}_{n=0}^{\mathrm{\infty}}\frac{{x}^{n1}}{(n1)!}$.
Tips for Differentiating Infinite Series:

TermbyTerm 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.