Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

b. Write the power series using summation notation.

f(x) = 1/x², a=1

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Identify the function and the center of the Taylor series: here, the function is \(f(x) = \frac{1}{x^2}\) and the series is centered at \(a = 1\).

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

Calculate the first four derivatives of \(f(x)\): - \(f(x) = x^{-2}\) - \(f'(x) = -2x^{-3}\) - \(f''(x) = 6x^{-4}\) - \(f^{(3)}(x) = -24x^{-5}\) Then evaluate each at \(x = 1\).

Write out the first four nonzero terms of the Taylor series using the formula: \[ T(x) \approx f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f^{(3)}(1)}{3!}(x-1)^3 \]

Express the Taylor series in summation notation: \[ T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(1)}{n!} (x - 1)^n \] where you can use the pattern found in the derivatives to write a general formula for \(f^{(n)}(1)\) if possible.

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Key Concepts

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

Taylor and Maclaurin Series Definition

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point a. The Maclaurin series is a special case centered at a = 0. Each term involves the nth derivative evaluated at a, multiplied by (x - a)^n and divided by n!. This expansion approximates the function near the point a.

Power Series and Summation Notation

A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Writing a Taylor series in summation notation compactly expresses all terms using an index n, making it easier to analyze and manipulate. Understanding this notation is essential for representing and working with series efficiently.

Interval of Convergence

The interval of convergence is the set of x-values for which the Taylor series converges to the function. It depends on the function and the center a. Determining this interval often involves applying convergence tests like the ratio test. Knowing the interval ensures the series accurately represents the function within that range.

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