Q. 7 - Calculus

Question

What is meant by the interval of convergence for a power series in $x-x_0$? How is the interval of convergence determined? If a power series in $x-x_0$ has a nontrivial interval of convergence, what types of intervals are possible. 

Step-by-Step Solution

Answer

Ans:  The interval of convergence of the power series in $x-x_0$ means the interval of values of x for which the series converges.  the interval of convergence may have one of the forms shown as $\left[-x_0,x_0\right],\left(-x_0,x_0\right),\left[-x_0,x_0\right)\text{ and }\left(-x_0,x_0\right]$

Also, we may have the case where a power series converge at any real value of $x$ . Hence, in this case, we may say that the interval of convergence of a power series is $R$.

Step 1. Given information.

given,

        $x-x_0$

Step 2. The interval of convergence of the power series in x - x 0 means the interval of values of x for which the series converges.

   So, when a power series in $x-x_0$ converges at a point $x_0$, then we can say that the interval of convergence for the series is trivial.

Step 3. It is often best to use the ratio test for absolute convergence to find those values for which lim k → ∞   a k + 1 a k x − x 0 < 1

It may be the case that a power series in $x-x_0$ converges at either $x_0$ or $-x_0$, or at both $x_0$ or $-x_0$, at neither of these points, or at just one of these points. Therefore, the interval of convergence may have one of the forms shown below

    $\left[-x_0,x_0\right],\left(-x_0,x_0\right),\left[-x_0,x_0\right)\text{ and }\left(-x_0,x_0\right]$

Step 4. Now,

Also, we may have the case where a power series converge at any real value of $x$ Hence, in this case, we may say that the interval of convergence of a power series is $R$.