% question 1

S(1) = 2/3;

for i = 2:17
    S(i) = S(i-1) + (-1)^i / (2*i+1);
end

figure(1);
plot(S, 0*S, 'ro', [.5 1], [0 0], 'g-');

% question 2

i = 1;
S_i = 2/3;

eps = 10^-6;

while abs(S_i - pi/4) >= eps
    i = i + 1;
    S_i = S_i + (-1)^i / (2*i + 1);
end

i

% question 3

i = 1;
S_i = 2/3;
prev = 1;
sigma = (S_i + prev) / 2;

eps = 10^-6;

while abs(sigma - pi/4) >= eps
    prev = S_i;
    i = i + 1;
    S_i = S_i + (-1)^i / (2*i+1);
    sigma = (S_i + prev) / 2;
end

i

% question 4

X(1) = 4;
Y(1) = 0;

for i = 1:3
    X(2*i) = X(2*i-1);
    Y(2*i) = X(2*i)^2 - 2;
    X(2*i+1) = X(2*i) - (X(2*i)^2 - 2) / (2*X(2*i));
    Y(2*i+1) = 0;
end

t = linspace(1, 5, 1000);

figure(2);
plot(X, Y, 'ro-', t, 0*t, 'g-', t, t.^2 - 2, 'b-');