$\text { Use the series } \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 1 } \text { for } f ( x ) = \arctan x \text { to approximate the value of }$ $\arctan \frac { 1 } { 8 }$ using $R _ { N } \leq 0.001$. Round your answer to three decimal places.
A) $0.125$
B) $0.111$
C) $0.143$
D) $0.100$
E) $0.133$

Question

Quizplus · Accepted Answer

The alternating series estimation theorem guarantees that the error is less than the first neglected term. So, $$R_N\le a_{N+1}=\frac{(-1)^{N+1}x^{2(N+1)+1}}{2(N+1)+1}=\frac{(-1)^{N+1}x^{2N+3}}{2N+3}$$For $R_N\le 0.001$, we have$$\frac{(-1)^{N+1}x^{2N+3}}{2N+3}\le 0.001$$$$\Rightarrow \frac{x^{2N+3}}{2N+3}\le 0.001$$$$\Rightarrow x^{2N+3}\le 0.001(2N+3)$$$$\Rightarrow x^{2N+3}\le 0.002N+0.003$$$$\Rightarrow x^{2N+3}\le 0.002N+0.003$$$$\Rightarrow x^{2N+3}\le 0.002(8)+0.003=0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{2N+3}\le 0.019$$$$\Rightarrow x^{

arctan⁡18\arctan \frac { 1 } { 8 }arctan81​ Using RN≤0.001R _ { N } \leq 0.001RN​≤0.001

$\arctan \frac { 1 } { 8 }$ Using $R _ { N } \leq 0.001$