Homework 7Integration and ApproximationDue 4/23 in recitationDirections:Solve all problems. You must show your work for full credit.Power series1. Find the radius of convergence of∞Xn=0x3n3n!.2. Without using Taylor’s formulacn=f(n)(a)n!, find the power series centered atafor the functions below.(a)11-2x,a= 0(b) sin2x(compute only the first 3 nonzero terms)3. Using Taylor’s formula, find the Maclaurin series of cosh(x)4. Find limx→0sin(x)-sinh(x)x3Differential equations5. Find a solution to the differential equations below.(a)xy0+y=y2,y(1) =-1.(b)dvdt=-v+ 10,v(0) = 2.(c)dxdt= (2-x)(1-x),x(0) = 0.6. A population is modeled by the differential equationdPdt= 2P1-P1000.(a) For what values ofPis the population increasing? For what values is it decreasing? You onlyneed to considerP≥0.(b) FindtwoconstantsP*so thatP(t) =P*is a solution. These calledequilibria.(c) This is a separable differential equation! Solve it with initial dataP(0) = 100.