Question

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)

Accepted Answer

Recall that the Taylor series expansion for $$ 	an^{-1}(x) $$ centered at $$ x=0  ($$also known as the Maclaurin series) is given by the infinite series:

$$ 
	an^{-1}(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots 
$$  
This series converges for $$ |x| \leq 1 . $$Since we want to approximate $$ 	an^{-1}(1/2) $$, substitute $$ x = \frac{1}{2} $$ into the series:

$$ 
	an^{-1}\left(\frac{1}{2}ight) = \frac{1}{2} - \frac{\left(\frac{1}{2}ight)^3}{3} + \frac{\left(\frac{1}{2}ight)^5}{5} - \frac{\left(\frac{1}{2}ight)^7}{7} + \cdots 
$$  
This gives the infinite series representation for $$ 	an^{-1}(1/2) . $$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: $$ - \frac{(1/2)^7}{7} = - \frac{1/128}{7} $$ Write the partial sum of the first four nonzero terms as:

$$ 
	an^{-1}\left(\frac{1}{2}ight) \approx \frac{1}{2} - \frac{1}{24} + \frac{1}{160} - \frac{1}{896} 
$$  
This expression represents the approximation using the first four terms of the Taylor series. To summarize, the steps to find the first four nonzero terms are:  
- Use the Maclaurin series for $$ 	an^{-1}(x) .  
- $$Substitute $$ x = \frac{1}{2} .  
- $$Write out the first four terms explicitly with powers and denominators.  
- Express the partial sum as the approximation of $$ 	an^{-1}(1/2) .$$

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)

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