Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = tan⁻¹(4x), a = 0

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Recall that the Taylor series of a function \( f(x) \) centered at \( a \) is given by \( f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \). Here, \( a = 0 \), so the series is centered at zero (Maclaurin series).

Identify the function: \( f(x) = \tan^{-1}(4x) \). We can use the known Maclaurin series for \( \tan^{-1}(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} \) for \( |x| < 1 \).

Substitute \( 4x \) in place of \( x \) in the series to get \( \tan^{-1}(4x) = \sum_{n=0}^\infty (-1)^n \frac{(4x)^{2n+1}}{2n+1} = \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1} x^{2n+1}}{2n+1} \).

Write out the first three nonzero terms by plugging in \( n = 0, 1, 2 \): - For \( n=0 \): \( \frac{4^1 x^1}{1} = 4x \) - For \( n=1 \): \( - \frac{4^3 x^3}{3} = - \frac{64 x^3}{3} \) - For \( n=2 \): \( \frac{4^5 x^5}{5} = \frac{1024 x^5}{5} \).

Finally, write the Taylor series in summation notation as \( \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1}}{2n+1} x^{2n+1} \).

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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. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This allows approximation of functions near the center point.

Derivatives of Inverse Trigonometric Functions

Understanding how to differentiate inverse trigonometric functions like arctan(x) is essential. For example, the derivative of arctan(x) is 1/(1 + x^2). When the function is arctan(4x), the chain rule applies, multiplying by the derivative of 4x, which is 4.

Summation Notation for Series

Summation notation concisely expresses infinite series using the sigma symbol (∑). It includes an index of summation, limits, and a general term formula. Writing the Taylor series in summation form captures all terms compactly and highlights the pattern in coefficients and powers.

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.ƒ(x) = tan⁻¹(4x), a = 0