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Hi , I`m Jack , I will be helping you today Can you please send me a reply when you are ready to proceed ?
Still get the audio if not the picture ?
Is the screen just blank with some background light or completely dark like when the tv is off ?If you`re not sure then darken the room and turn the tv off and on again while you observe the screen .
If completely dark , do you get a brief flash of light on the screen when you first turn it on ?
Never mind the last question , in any case I can tell you that this model has problems with its backlight inverter circuits , namely the high voltage transformer on it .
If you are into this you could buy these transformers at the link below and replace /solder them on your board but it may be easier to simply replace that board .
You can see the various versions at that link , not in stock there but first to match yours and then with the correct part number it could be found.
Here is that link ; http://www.shopjimmy.com/catalogsearch/result/?q=ln40b610
Other wise you are looking at a couple hundred dollars repaired in a shop and for a 40" tv I`m not sure its worth the repair .
I will have to step out for a bit but If that answers your question please do not forget to rate by clicking on the star icons at the top, then click on Submit so i get paid for my time helping you .Also note that it will not close this post , you can come back here at any time if you have questions.