New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x 0 and successively defines a sequence z n + 1 = x n − f ( x n ) f ' ( x n ) . For the given choice of f and x 0 . write out the formula for x n + 1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x n agrees with x up to four decimal places. 59. [ T ] A lake initially contains 2000 fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by 6% each month. However, factoring in all causes, 150 fish ate lost each month. a. Explain why the fish population after ii months is modeled by P n = 1 .06P n — — 150 with P 0 = 2000. b. How many fish will be in the pond after one year?
New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x 0 and successively defines a sequence z n + 1 = x n − f ( x n ) f ' ( x n ) . For the given choice of f and x 0 . write out the formula for x n + 1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x n agrees with x up to four decimal places. 59. [ T ] A lake initially contains 2000 fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by 6% each month. However, factoring in all causes, 150 fish ate lost each month. a. Explain why the fish population after ii months is modeled by P n = 1 .06P n — — 150 with P 0 = 2000. b. How many fish will be in the pond after one year?
New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence
z
n
+
1
=
x
n
−
f
(
x
n
)
f
'
(
x
n
)
. For the given choice of f and x0. write out the formula for
x
n
+
1
. If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places.
59. [T] A lake initially contains 2000 fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by 6% each month. However, factoring in all causes, 150 fish ate lost each month.
a. Explain why the fish population after ii months is modeled by Pn= 1 .06P n— — 150 with P0= 2000.
b. How many fish will be in the pond after one year?
To find the root of the function f(x), write down the mapping function, g(x),
for Newton’s method. Using x0 = π/3 as the initial guess, perform the first four
iterations using Newton’s method. Also calculate the ratio |xn/xn−1| at every step.
What does this ratio denote and what does its limiting value tell you about the
convergence rate of Newton’s method?
Find the radius of convergence and interval of convergence of
Σ 2 to infinity ((x+2)^n)/(2^n)(ln n)
∑∞ ((−1)^n−1(1−n)) /(3n−n^2)
n=1
Find if the series converges or diverges!
Find the limit if it converges.
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