The power series expansion of f(z) = 1/(1+2z) about the point P=0 is given by ∑ (-1)ⁿ (2z)ⁿ for n=0 to ∞, with a disc of convergence for |z| < 1/2.
To find the power series expansion of the function f(z) = 1/(1+2z) around the point P = 0, we can use the geometric series. The geometric series formula (1+x)⁻¹ = 1 - x + x² - x³ + ..., converges when |x| < 1. In our case, we can treat 2z as x and use the formula to get the power series:
f(z) = 1 - 2z + (2z)²- (2z)³ + ... = ∑ (-1)ⁿ (2z)ⁿ for n=0 to ∞.
The disc of convergence for this series is found by the inequality |2z| < 1, which gives us |z| < 1/2. Therefore, the radius of convergence is 1/2, and the disc of convergence is the set of points z such that |z| < 1/2.