The series (4x-3)+(4x-3)^2+(4x-3)^3................................is given for which values of X will the series converge??
To determine the values of x for which the series converges, we need to analyze the behavior of the terms as x varies. Let's break down the given series and examine its individual terms:
(4x-3): This is the first term of the series and contains a linear expression in x. The behavior of this term depends on the value of x but does not impact the convergence of the entire series.
(4x-3)^2: This is the second term of the series and contains a quadratic expression in x. The squaring operation ensures that this term is always non-negative, regardless of the value of x. Therefore, the convergence of the series does not depend on this term.
(4x-3)^3: This is the third term of the series and contains a cubic expression in x. Similar to the second term, cubing the quadratic expression ensures that this term is always non-negative.
Based on the analysis of the individual terms, we can conclude that the convergence of the series is determined solely by the convergence of the linear term (4x-3).
For a series to converge, the limit of its terms as x approaches a certain value must exist and be finite. In the case of the linear term (4x-3), the limit exists for all values of x.
Therefore, this series will converge for all real values of x.