The function $g$ has derivatives of all orders for all real numbers. The Maclaurin series for $g$ is given by $g ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 2 e ^ { n } + 3 }$ on its interval of convergence. (a) State the conditions necessary to use the integral test to determine convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$. Use the integral test to show that $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ converges. (b) Use the limit comparison test with the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ to show that the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ converges absolutely. (c) Determine the radius of convergence of the Maclaurin series for $g$. (d) The first two terms of the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ are used to approximate $g ( 1 )$. Use the alternating series error bound to determine an upper bound on the error of the approximation.
The function $g$ has derivatives of all orders for all real numbers. The Maclaurin series for $g$ is given by $g ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 2 e ^ { n } + 3 }$ on its interval of convergence.\\
(a) State the conditions necessary to use the integral test to determine convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$. Use the integral test to show that $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ converges.\\
(b) Use the limit comparison test with the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ to show that the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ converges absolutely.\\
(c) Determine the radius of convergence of the Maclaurin series for $g$.\\
(d) The first two terms of the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ are used to approximate $g ( 1 )$. Use the alternating series error bound to determine an upper bound on the error of the approximation.