(4) 0,42 (c) Determine the values of \( x \) for which each of the following geometric series converges: \( \begin{array}{ll}\text { (1) } x+2 x^{2}+4 x^{3}+\ldots & \text { (2) } 1-x+x^{2}-\ldots \\ \text { (3) }(x+1)+(x+1)^{2}+(x+1)^{3}+\ldots & \text { (4) } 2(1-x)+2(1-x)^{2}+2(1-x)^{3}+\ldots \\ \text { (5) }(2 x-5)+(2 x-5)^{2}+(2 x-5)^{3}+\ldots & \text { (6) } x(7-2 x)^{4}-x(7-2 x)^{5}+x(7-2 x)^{6}- \\ \text { (7) }(3 x-2)^{3}-(3 x-2)^{4}+(3 x-2)^{5}-\ldots & \text { (8) }(2-4 x)+2(2 x-1)^{2}-2(2 x-1)^{3}+\ldots\end{array} \)

To determine the values of \( x \) for which each of the given geometric series converges, we need to recall the condition for convergence of a geometric series. A geometric series of the form \( a + ar + ar^2 + ar^3 + \ldots \) converges if the common ratio \( r \) satisfies \( |r| < 1 \). Let's analyze each series one by one: ### (1) \( x + 2x^2 + 4x^3 + \ldots \) This series can be rewritten as: \[ x + 2x^2 + 4x^3 + \ldots = x(1 + 2x + 4x^2 + \ldots) \] The common ratio \( r = 2x \). For convergence: \[ |2x| < 1 \implies |x| < \frac{1}{2} \] ### (2) \( 1 - x + x^2 - \ldots \) This series can be rewritten as: \[ 1 - x + x^2 - \ldots = \frac{1}{1+x} \quad \text{(geometric series with } r = -x\text{)} \] For convergence: \[ |-x| < 1 \implies |x| < 1 \] ### (3) \( (x+1) + (x+1)^2 + (x+1)^3 + \ldots \) This series can be rewritten as: \[ (x+1) + (x+1)^2 + (x+1)^3 + \ldots \] The common ratio \( r = (x+1) \). For convergence: \[ |x+1| < 1 \implies -1 < x + 1 < 1 \implies -2 < x < 0 \] ### (4) \( 2(1-x) + 2(1-x)^2 + 2(1-x)^3 + \ldots \) This series can be rewritten as: \[ 2(1-x) + 2(1-x)^2 + 2(1-x)^3 + \ldots \] The common ratio \( r = (1-x) \). For convergence: \[ |1-x| < 1 \implies 0 < x < 2 \] ### (5) \( (2x-5) + (2x-5)^2 + (2x-5)^3 + \ldots \) This series can be rewritten as: \[ (2x-5) + (2x-5)^2 + (2x-5)^3 + \ldots \] The common ratio \( r = (2x-5) \). For convergence: \[ |2x-5| < 1 \implies 4 < 2x < 6 \implies 2 < x < 3 \] ### (6) \( x(7-2x)^4 - x(7-2x)^5 + x(7-2x)^6 - \ldots \) This series can be rewritten as: \[ x(7-2x)^4(1 - (7-2x) + (7-2x)^2 - \ldots) \] The common ratio \( r = -(7-2x) \). For convergence: \[ |-(7-2x)| < 1 \implies |7-2x| < 1 \implies 6 < 2x < 8 \implies 3 < x < 4 \] ### (7) \( (3x-2)^3 - (3x-2)^4 + (3x-2)^5 - \ldots \) This series can be rewritten as: \[ (3x-2)^3(1 - (3x-2) + (3x-2)^2 - \ldots) \] The common ratio \( r = -(3x-2) \). For convergence: \[ |-(3x-2)| < 1 \implies |3x-2| < 1 \implies 1 < 3x < 3 \implies \frac{1}{3} < x < 1 \] ### (8) \( (2-4x) + 2(2x-1)^2 - 2(2x-1)^3 + \ldots \) This series can be rewritten as: \[ (2-4x) + 2(2x-1)^2 - 2(2x-1)^3 + \ldots \] The common ratio \( r = -(2x-1) \). For convergence: \[ |-(2x-1)| < 1 \implies |2x-1| < 1 \implies 0 < 2x < 2 \implies 0 < x < 1 \] ### Summary of Convergence Conditions 1. \( |x| < \frac{1}{2} \) 2. \( |x| < 1 \) 3. \( -2 < x < 0 \) 4. \( 0 < x < 2 \) 5. \( 2 < x < 3 \) 6. \( 3 < x < 4 \) 7. \( \frac{1}{3} < x < 1 \) 8. \( 0 < x < 1 \) These intervals provide the values of \( x \) for which each series converges.