While the action of pinging/checking links is rather simple, access to the server that hosts your website can and does vary a lot, depending on the software that your hoster uses to give you access to your website.
Very few hosters give you command-line access directly, (using TelNet/TTY client, like 'Putty') but that is usually the simplest way for their customer to execute commands like: ping https://pagead2.googlesyndication.com/
Most hosters give you an online version of a TelNet/TTY client nowadays. Which is usually a lot less advanced than a dedicated client and slower too. But when you are in a pinch, it suffices.
When you don't get any response from your ping command, it's highly likely that this particular domain is not available anymore and you'll need to find out which Google domain is currently hosting those scripts/ads.
If you do get a valid response, then you could use a browser on the hoster's server and see if you get the content from the file 'show_ads.js' in the link https://pagead2.goog...m/pagead/show_ads.js
If you get valid content there, the file/folder structure at Google hasn't changed, so you can be certain the your problem is not this particular link. However, if you don't get any content or garbage content, you'll have found the (first?) error on your end. Unfortunately, you must repeat this for every link that your website generates. It won't be too difficult of a job, but time-consuming it definitely will be.
Did you try to see the ads in your site using only your favorite browser inside your own network? If so, go and use a computer/tablet/laptop that you do not own in a network that isn't yours. With permission, of course.
If ads de show up on that device, you can be sure that you have a local problem (as in: your computer/browser is blocking your ads and you'll need to go revise all extensions/configuration options on your own computer.
But if the ads also not show on the other computer, it is more likely that your problem is related to your hoster and/or your website configuration than anything else. And that analysis needs a very different approach. How different? That depends on your hoster and the software they use to manage their hosting equipment.
All of the above is more or less manual labor and I'm sure that a developer extension, making most of the above a lot easier, does exists for your favorite browser. Yet going through the motions manually will give you a troubleshooting mindset, which will help you solving future problems.
Logging files give you better insights too. Perhaps even more than the manual stuff above, but I understand that you are S.O.L. on that front, unfortunately.