Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.
tan ⁻¹ (1/2)

Verified step by step guidance

1. Recall that the Taylor series expansion for \( \tan^{-1}(x) \) centered at \( x=0 \) (also known as the Maclaurin series) is given by the infinite series: \[ \tan^{-1}(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \] This series converges for \( |x| \leq 1 \).
2. Since we want to approximate \( \tan^{-1}(1/2) \), substitute \( x = \frac{1}{2} \) into the series: \[ \tan^{-1}\left(\frac{1}{2}\right) = \frac{1}{2} - \frac{\left(\frac{1}{2}\right)^3}{3} + \frac{\left(\frac{1}{2}\right)^5}{5} - \frac{\left(\frac{1}{2}\right)^7}{7} + \cdots \] This gives the infinite series representation for \( \tan^{-1}(1/2) \).
3. Identify the first four nonzero terms explicitly by writing each term with the powers and denominators clearly: 1st term: \( \frac{1}{2} \) 2nd term: \( - \frac{(1/2)^3}{3} = - \frac{1/8}{3} \) 3rd term: \( + \frac{(1/2)^5}{5} = + \frac{1/32}{5} \) 4th term: \( - \frac{(1/2)^7}{7} = - \frac{1/128}{7} \)
4. Write the partial sum of the first four nonzero terms as: \[ \tan^{-1}\left(\frac{1}{2}\right) \approx \frac{1}{2} - \frac{1}{24} + \frac{1}{160} - \frac{1}{896} \] This expression represents the approximation using the first four terms of the Taylor series.
5. To summarize, the steps to find the first four nonzero terms are: - Use the Maclaurin series for \( \tan^{-1}(x) \). - Substitute \( x = \frac{1}{2} \). - Write out the first four terms explicitly with powers and denominators. - Express the partial sum as the approximation of \( \tan^{-1}(1/2) \).

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It approximates functions near that point, allowing complex functions to be expressed as polynomials. For example, the Taylor series of arctan(x) at x=0 is used to approximate values like arctan(1/2).

Maclaurin Series for arctan(x)

The Maclaurin series is a Taylor series centered at zero. For arctan(x), it is given by x - x^3/3 + x^5/5 - x^7/7 + ... . This alternating series converges for |x| ≤ 1 and is used to approximate arctan values by substituting x with the desired number, such as 1/2.

Finding Nonzero Terms in a Series

When approximating a function using a Taylor series, identifying the first few nonzero terms involves calculating terms until four nonzero coefficients appear. This ensures a more accurate approximation. For arctan(1/2), the first four nonzero terms come from substituting x=1/2 into the first four terms of its Maclaurin series.