Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362600 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0499 1212040 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0499 1212040 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1164 4040176 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1177 4067936 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1183 4085648 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1184 4102656 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1191 4110392 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1191 4110392 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1918 4354304 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1933 4371704 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1938 4392632 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2024 4465168 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2024 4465424 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

## Proof of the Radius of Convergence Theorem

For a given power series,precisely one of the following may happen

1. the series converges only for

2. the series converges for all

3. there is some real numbersuch thatconverges and converges absolutely ifanddiverges if

The proof rests on the claim that if a power series is convergent for someon a circle with centre the origin, then it is absolutely convergent for all points inside the circle.

Note that ifis convergent thenso that for some

Hence forand the series converges so the seriesconverges by the comparison test.

Suppose that neither 1 or 2 above hold. Consider the set

Since 1 does not hold there is somesuch thatis convergent and hence the setis not empty. Moreover,is an interval (and not a set of discrete points) sinceandthenSince 2 does not hold there is somesuch thatis divergent and henceis divergent. Thusand so the interval of convergence has a finite, non zero, right hand end point(which may or may not be in).

Ifthenis convergent henceis absolutely convergent.

On the other hand, ifthen we can chooseto satisfythenis divergent (since) henceis divergent.