Taylor series

b. Write the power series using summation notation.

f(x) = 1/x, a = 1

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Recall that the Taylor series of a function \(f(x)\) centered at \(a\) is given by the formula: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\] where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).

Identify the function and center: here, \(f(x) = \frac{1}{x}\) and the center is \(a = 1\).

Compute the derivatives of \(f(x)\) and evaluate them at \(x = 1\): - \(f(x) = x^{-1}\) - \(f'(x) = -x^{-2}\) - \(f''(x) = 2x^{-3}\) - \(f^{(3)}(x) = -6x^{-4}\) - and so on, noticing the pattern in the derivatives.

Evaluate each derivative at \(x = 1\) to find \(f^{(n)}(1)\), which will simplify the coefficients in the series.

Write the Taylor series in summation notation by substituting \(f^{(n)}(1)\) and \(n!\) into the formula: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!} (x - 1)^n.\] Express the general term explicitly using the pattern found for \(f^{(n)}(1)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function f(x) centered at a point a, the series is given by f(x) = Σ (f⁽ⁿ⁾(a)/n!) (x - a)ⁿ, where f⁽ⁿ⁾(a) is the nth derivative evaluated at a.

Derivatives of the Function

To write the Taylor series, you need to find the derivatives of the function f(x) = 1/x at the point a = 1. Each derivative provides the coefficients for the series terms, and recognizing the pattern in these derivatives helps express the series in a general summation form.

Summation Notation for Power Series

Summation notation compactly expresses infinite series using the sigma symbol (Σ). Writing the Taylor series in summation form involves identifying the general term of the series and representing it as Σ from n=0 to ∞ of the nth term, which includes the derivative coefficient, factorial denominator, and (x - a)ⁿ factor.

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