The function $f$ is defined by the power series $$f ( x ) = 1 + ( x + 1 ) + ( x + 1 ) ^ { 2 } + \cdots + ( x + 1 ) ^ { n } + \cdots = \sum _ { n = 0 } ^ { \infty } ( x + 1 ) ^ { n }$$ for all real numbers $x$ for which the series converges. (a) Find the interval of convergence of the power series for $f$. Justify your answer. (b) The power series above is the Taylor series for $f$ about $x = - 1$. Find the sum of the series for $f$. (c) Let $g$ be the function defined by $g ( x ) = \int _ { - 1 } ^ { x } f ( t ) d t$. Find the value of $g \left( - \frac { 1 } { 2 } \right)$, if it exists, or explain why $g \left( - \frac { 1 } { 2 } \right)$ cannot be determined. (d) Let $h$ be the function defined by $h ( x ) = f \left( x ^ { 2 } - 1 \right)$. Find the first three nonzero terms and the general term of the Taylor series for $h$ about $x = 0$, and find the value of $h \left( \frac { 1 } { 2 } \right)$.
The function $f$ is defined by the power series
$$f ( x ) = 1 + ( x + 1 ) + ( x + 1 ) ^ { 2 } + \cdots + ( x + 1 ) ^ { n } + \cdots = \sum _ { n = 0 } ^ { \infty } ( x + 1 ) ^ { n }$$
for all real numbers $x$ for which the series converges.
(a) Find the interval of convergence of the power series for $f$. Justify your answer.
(b) The power series above is the Taylor series for $f$ about $x = - 1$. Find the sum of the series for $f$.
(c) Let $g$ be the function defined by $g ( x ) = \int _ { - 1 } ^ { x } f ( t ) d t$. Find the value of $g \left( - \frac { 1 } { 2 } \right)$, if it exists, or explain why $g \left( - \frac { 1 } { 2 } \right)$ cannot be determined.
(d) Let $h$ be the function defined by $h ( x ) = f \left( x ^ { 2 } - 1 \right)$. Find the first three nonzero terms and the general term of the Taylor series for $h$ about $x = 0$, and find the value of $h \left( \frac { 1 } { 2 } \right)$.