The function is defined for near by the power series g(x)= _ n=0 ^ (3x)^ n (n+1)2^ n
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
The function g is defined for x near 0 by the power series
g(x)=n=0∑∞(n+1)2n(3x)n.
Assume all series manipulations below occur inside the interval of convergence.
(a) Determine the radius of convergence of the power series defining g(x).
(b) Differentiate the series term-by-term to obtain a power series representation for g′(x) valid on the interval of convergence.
(c) Express g′(x) in closed form as an elementary function for all x in the interval found in part (a).
(d) Using an antiderivative of the function found in part (c), evaluate g(0.4) exactly.
(e) Instead of the exact value, suppose g(0.4) is approximated by truncating the original series after N terms. Determine the smallest integer N for which the resulting approximation differs from the exact value by less than 10−7, and compute the corresponding partial-sum approximation. Provide a justification that your choice of N meets the required error bound.