If you are using jersey 2.x then you need different configuration in web.xml as servlet class is change in it. you can update your web.xml with following configuration.
<servlet>
<servlet-name>myrest</servlet-name>
<servlet-class>org.glassfish.jersey.servlet.ServletContainer</servlet-class>
<init-param>
<param-name>jersey.config.server.provider.packages</param-name>
<param-value>your.package.path</param-value>
</init-param>
<init-param>
<param-name>unit:WidgetPU</param-name>
<param-value>persistence/widget</param-value>
</init-param>
<load-on-startup>1</load-on-startup>
</servlet>
<servlet-mapping>
<servlet-name>myrest</servlet-name>
<url-pattern>/rest/*</url-pattern>
</servlet-mapping>