Interval of Convergence Calculator: Radius & Series Tester

In advanced mathematics, particularly in calculus and analysis, power series are used to represent complex functions (like trigonometric, exponential, or logarithmic curves) as infinite polynomials. For a power series to represent a function, it must converge to a finite value. The set of all $x$ values for which a power series converges is called the **Interval of Convergence**. ### Securing the Radius of Convergence The foundation of analyzing convergence involves finding the **Radius of Convergence** ($R$): - If $R = 0$, the series converges exclusively at its center point $x = c$. - If $R = infty$, the series converges absolutely for all real numbers. - If $0 < R < infty$, the series converges inside the bounds $c-R < x < c+R$. To solve for $R$, mathematicians apply the **Ratio Test**: $$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$ Setting $L < 1$ determines the inequality boundaries, establishing $R = 1/L$. ### Testing the Corner Boundaries The Ratio Test is silent on endpoint convergence. We must analyze the boundary conditions independently: - Substitute $x = c - R$ back into the original series and evaluate (using Alternating Series, P-Series, or Integral tests). - Substitute $x = c + R$ back into the series and evaluate. These steps determine whether the intervals use parenthetical open bounds '(' or bracketed closed bounds ']'. If the series converges at $c-R$, the interval is closed on the left $[c-R, c+R)$.