A particle moves along a line so that its velocity v (in m/s) for ∣t∣<1 is modeled by the power series
v(t)=n=0∑∞2n+1(−3)nt2n+1.
(a) Determine the radius of convergence of the series for v.
(b) The particle is at the origin when t=0. Obtain a power series for the position function s(t) valid on the interval of convergence and satisfying s(0)=0.
(c) Use the series from part (b) to approximate the displacement of the particle over the time interval 0≤t≤0.2, writing your answer to five decimal places.
(d) Explain why the approximation obtained in part (c) is guaranteed to agree with the exact displacement to within 10−6 meters.
a. Part (a)
b. Part (b)
c. Part (c)
d. Part (d)