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: 0.0000 362456 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0512 1211800 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0512 1211800 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1199 4127352 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1213 4154936 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1221 4172648 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1222 4203168 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1232 4225888 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1232 4225888 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1931 4897312 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1966 5089464 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1971 5110392 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2081 5305184 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2081 5305440 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

The Harmonic Series

The Harmonic series is the infinite series whose sum is given by

The harmonic series diverges. This is quite easy to prove.

Thusis unbounded sois divergent.

In factis in many cases divergent for manyfor example if

S is the set of prime numbers

S is the set of multiples of any number.

The series diverges very slowly. For example, the sum of the firstterms is less than 100. This is because the partial sums of the series have logarithmic growth:

as can easy be proved using the integral test.