Consider a branching process with offspring distribution {bk}k> 0 and assume 0 < bo < 1. suppose the evolution is altered as follows. start with xo = 1, a single individual. if the next generation is not empty (that is, if x1 > 0), they invite an immigrant to join them (that is, x1 is replaced by x1 + 1). after that the evolution continues according to the usual rule. in particular, no further immigration takes place. find the extinction probability î of this new process. express it in terms of bo and the original extinction probability a. do we have ô < a for all 0 < a < 1, in other words, does the immigrant strictly improve chances of survival for all values 0 < a < 1? is it possible that û

Proceed as in example 3 in section 6.1 to rewrite the given expression using a single power series whose general term involves xk. ∞ n(n − 1)cnxn − 2 n = 2 − 4 ∞ ncnxn n = 1 + ∞ cnxn n = 0

Asystem of linear equations with more equations than unknowns is sometimes called an overdetermined system. can such a system be consistent? illustrate your answer with a specific system of three equations in two unknowns. choose the correct answer below. a. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 6 b. no, overdetermined systems cannot be consistent because there are fewer free variables than equations. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 12 c. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 8 d. no, overdetermined systems cannot be consistent because there are no free variables. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 24

Solve for x where 0 equation: cot^2x+2cotx+1=0