What is the correct sizing for a cropped jpeg to display on large tv?

Because you want to “get a great closeup of an animal and crop the rest,” it sounds like you should not use Image Size or an automated solution. You probably want to quickly crop each image down to the best part. If that’s correct, then you only need to do one thing: When the Crop tool is selected, enter the TV pixel dimensions into the options bar. If the residence has an older HDTV, use 1920px by 1080px as shown, but if they have a 4K TV then use 3840px by 2160px as suggested earlier. (For a TV or projector, it isn't necessary to enter a px/in resolution.)

As you drag the Crop tool to get the crop you want, the settings you entered in the options bar will keep the Crop tool locked to the correct proportions. When you commit the crop by clicking the check mark in the options bar (or by pressing Enter or Return), Photoshop will set the cropped image to the pixel dimensions in the options bar, no matter what the original image size was. Very quick and reliable.

When you’re done with all the images, click Clear in the options bar so that those settings don’t constrain future crops on other images.

To fill the TV screen images must have a 16:9  wide language aspect ratio. You can center crop most image with a normal landscape camera's aspect ratio ie 3:2. 4:3, 5:4 to a 6:9 aspect ratio and have an acceptable image composition.  However Portrait originated camera images  will not have an acceptable composition cropped to a 16:9 wide landscape aspect ratio. A simple three step action the uses two Photoshop Plug-in scripts could be batch to center crop all your landscape images to a 6:9 aspect ratio the size then size the image to the number of pixels your HDTV has.  720p 1280pxBy720px, 1080p 1920pxBy1080px, 4K 3840pxBy2160px, 8K 7680pxBy4320pc

Step 1 AspecrRatioSelection 16 9

Step 2 Image Crop

Step 3 Fit Image TV width TV Height

Hi
Video works in pixels.
The standard for HD television is 1920 x 1080  The newer 4K UHD televisions are 3840 x 2160 
As Gene said - forget about ppi - that is for printing
Dave

I've never tried, but until better answers come along, see this article:

https://www.scantips.com/basics1f.html

and from the forums I found this: https://community.adobe.com/t5/Photoshop/Display-a-Picture-on-a-TV/td-p/7426931

Dpi/ppi is meaningless on a screen, don't worry about it.

You need to start by finding out the resolution of the screen and remember to set your Ps file's res to 72dpi.