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Taylor series are a mathematical tool for representing functions as infinite sums of their derivatives at a point. They are crucial for approximating functions, understanding convergence properties, and simplifying evaluation, differentiation, and integration in analysis. Examples include the exponential function and geometric series, with applications across various scientific domains.

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## Definition of Taylor Series

### Representation of functions as infinite sums

Taylor series represent functions as infinite sums of terms derived from the function's derivatives at a single point

### Structure of Taylor series

Power series centered at a point

Each term of the Taylor series incorporates a higher-order derivative of the function at a point, multiplied by the corresponding power of (x-a) and divided by the factorial of the term's order

Summation notation

The Taylor series can be expressed using summation notation, where the index of summation starts at 0 and extends to infinity

### Examples of Taylor series

Common functions such as the exponential and geometric series can be represented by their Taylor series

## Convergence properties of Taylor series

### Radius of convergence

The radius of convergence is the distance within which the Taylor series converges to the function it represents

### Interval of convergence

The interval of convergence defines the set of x values for which the Taylor series converges to the function

### Truncated Taylor series

A truncated Taylor series, consisting of a finite number of terms, can be used to approximate a function with a quantifiable error

## Applications of Taylor series

### Alternative representations of functions

Taylor series provide alternative representations of functions that can simplify processes such as evaluation, differentiation, and integration

### Integration using Taylor series

Functions without elementary antiderivatives can often be integrated using their Taylor series expansions

### Versatility of Taylor series

Taylor series can be used to solve problems that are challenging to solve using standard analytical methods, making them a foundational concept in calculus and analysis