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 362584 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0543 1212024 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0543 1212024 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.2147 4127608 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.2161 4155288 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.2169 4173000 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.2171 4203520 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.2182 4226240 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.2182 4226240 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2793 4897328 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2831 5089480 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2837 5110408 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2954 5304872 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2954 5305128 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Series - Standard Expressions for Sums of Powers of Integers

There are several very useful expressions that expression the sum of powers of integers in terms of the range of summation.

Notice that the degree of the expression on the right hand side is one more than the terms of the left hand side, so that summing a polynomial sequence increases the degree by one.

The series is always finite so n is a finite number and the series has a finite number of terms to add.

There are no such simple expressions for sums of fractional or decimal powers.

The expressions above, and similar expressions, act as building blocks which enable expressions for the sum of any finite polynomial series to be formed.