how to update iMac 10.13.6 version

Your upgrade options depend totally on your iMac's sub-model and current macOS version, neither of which you have shared with us.

Both are quickly available from "About this Mac.. " under your Apple menu:

We cannot see, touch, nor remotely access your computer in this setting, so must rely on you for details.

Thanks

Hi,

The HP DeskJet 2700e All-in-One series appears to support the driverless AirPrint technology.

About AirPrint - Apple Support

An AirPrint-enabled printer is per definition compatible with macOS, and can be used without additional drivers/software, provided that the printer and the Mac computer are connected to the same network (Wi-Fi router).

The problem is that HP requires a higher Mac operating system than needed for AirPrint per se. The reason for this seems to be related to setup, where the HP Smart app for Mac is used, which needs macOS 10.14 Mojave or higher. Also, there are further requirements for an HP+ service (including an always active Internet connection).

It is interesting to note that the manual for the printer http://h10032.www1.hp.com/ctg/Manual/c07043348.pdf on page 46 says "The printer is compatible with Mac computers running macOS or OS X 10.7 and later and Apple AirPrint.", though.

Could a workaround possibly be to use an iPhone (iOS 13 or higher), iPad (iPadOS 13 or higher), Android (7.0) phone, or a Windows (10) computer just for the necessary setup procedure with HP Smart (and ink management?), and thereafter print via AirPrint under macOS 10.13 High Sierra on the Mac without extra software? I do not know. It may depend upon what kind of service (such as HP+) you have ordered when the printer was purchased. HP support should be able to answer these questions.

See also https://support.hp.com/us-en/document/ish_3872876-3834337-16

OK. This would mean that the HP printer and the iPhone are connected to the same Wi-Fi network (router). Correct?

Would it also be correct to assume that the Mac computer already is connected to that same Wi-Fi network? That is, you are probably using the Mac via the Wi-Fi router for Internet access.

If so, it ought to be possible to add the printer to the Mac (running macOS 10.13 High Sierra) using AirPrint only. See also this article (the information about macOS 10.14 Mojave should be valid for macOS 10.13 High Sierra as well):

Add a printer on Mac – Apple Support (UK)

A guide from Canon (!) shows how a printer can be added to use AirPrint:

https://www.canon.co.uk/support/consumer_products/apple_airprint_guide/#Pmac

If there is a problem with device discovery with AirPrint, the following power-cycle procedure may help (do not forget the router): Power off the Wi-Fi router, the HP printer, the iPhone, the Mac computer, and all other devices/computers. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two. Power on the iPhone. Power on the Mac computer. Finally, power on all other devices/computers.

No further OS upgrades are compatible with a pre-2012 Mac model except for the 2010 desktop Mac Pros.

(229419)

Be aware that Apple considers that model obsolete. No more security updates nor system. It's time to start saving up for a new iMac. The one you have has reached its limit.

That says this is a MacBook Pro 13-inch Late 2011. I hope that does not affect the responses you get here in the iMac forums.