Q. 16 - Calculus

Question

Is it possible for a power series to have $(0,\infty )$ as its interval converge? Explain your answer.

 

Step-by-Step Solution

Answer

If there is a positive real integer $\rho$, the series will therefore absolutely converge for every $x\in (-\rho ,\rho )$

Step 1. Given information.

Given, Is the interval convergence of a power series ever $\left(0,\infty \right)$

Step 2: Explanation

 It is impossible for a power series with an x value to have an interval of convergence of $(0,\infty )$

The reason for this is if the power series $\sum _{i=0}^{\infty }{a}_k{x}^k$ is the power series in x.

The series will therefore absolutely converge for every $x\in (-\rho ,\rho )$ if there is a positive real number $\rho$.