The Taylor Series is the expansion of a given function into the infinite sum of terms in which each sum has a large exponent. Each term in the series is generally the derivated of that function at that point. If the derivative is equal to zero at that point, the Taylor series is also a Maclaurin series.

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Who found the Taylor Series?

Brook Taylor was an English mathematician born in 1685 who in 1715, found a general method for constructing these series and hence had these series named after him. Many other mathematicians also worked on these series, including the Scottish mathematician and astronomer James Gregory with work on the series even dating back to the Indian mathematician the Madhava of Sangamagrama in the 14th century.

What about the Maclaurin series?

The Maclaurin series was named after an 18th-century professor in Edinburgh Colin Maclaurin, who found a special case of the Taylor series where the derivative were considered at 0.

What are the practical (real-life) applications of the Taylor Series?

The Taylor series is highly useful as we can use them to calculate the value of a function at every point if all the function's derivatives are known at a single point. The Taylor series can be used to:

• Evaluate definite integrals, especially where functions have no antiderivative that can be expressed in terms of familiar functions.
• Understand asymptomatic functions and their behaviour.
• Computing transcendental functions such as e^x, sinx, and cos x.
• Useful for approximating function values and improving approximation accuracy by adding more terms to a series.