Final answer:
The Taylor series for 1/(1 + x) around x_0 = 0 is 1 - x + x^2 - x^3 + ... and the radius of convergence is 1, which means the series converges for |x| < 1.
Step-by-step explanation:
The Taylor series for the function 1/(1 + x) around x₀ = 0 is found using the formula for a geometric series. The function can be written as (1 - (-x))-1, which resembles the sum of a geometric series with a ratio of -x. Therefore, the Taylor series expansion is:
- 1 - x + x2 - x3 + ... + (-1)nxn + ...
The radius of convergence for this series is determined using the ratio test, which can be applied to the series absolute values. It gives us the condition that |x| < 1 for the series to converge. Consequently, the radius of convergence is 1.