\( x \) values where it converges \( 1+(2 x+1)+(2 x+1)^{2}+\ldots \)

The series you've provided is: \[ 1 + (2x + 1) + (2x + 1)^2 + (2x + 1)^3 + \dots \] This is a **geometric series** where the first term \( a = 1 \) and the common ratio \( r = 2x + 1 \). ### Convergence of a Geometric Series A geometric series converges **if and only if** the absolute value of the common ratio is **less than 1**: \[ |r| < 1 \] Substituting \( r = 2x + 1 \): \[ |2x + 1| < 1 \] ### Solving for \( x \) Let's solve the inequality: \[ |2x + 1| < 1 \] This absolute value inequality translates to: \[ -1 < 2x + 1 < 1 \] Subtract 1 from all parts of the inequality: \[ -2 < 2x < 0 \] Divide all parts by 2: \[ -1 < x < 0 \] ### Conclusion The series **converges** for all real numbers \( x \) in the open interval: \[ -1 < x < 0 \] **Therefore, the series converges for all real numbers x between –1 and 0.**