Do you think pushing a product at the end of it's life cycle through customer service efforts would be beneficial?

A product at the end of its PLC (Product Life Cycle) is a candidate for a Harvesting Strategy where one systematically withdraws marketing support to the bare minimum to sustain sales for the core customers of the product. So customer service at this point will not be useful rather effort should be made to make it available in select core markets, channels and minimum maintenance support. The objective is to reduce costs and squeeze out the maximum profits so that the investment dollars are moved to profitable and/or fast growing products.

Not really since many products reaching end of LC are consumed by late adopters who are generally looking for good deals thus less loyal to a brand. If the product needs CS to be operated properly then I would rather imagine a web site that brings all information needed to the customer; it is easy very effective and not expensive.

I don't think it's useful, it's a tactic to make quick cash by squeezing everything possible out of a product that's becoming obsolete.  More importantly it demonstrates total disregard for the customer.  It's far more beneficial to focus in advance of the curve, to innovate and develop and become a brand to watch (and buy) because your products are anticipating customer needs.  

every market's decision must to be based on the balance of demand and supply where the consumers and the producers were winners. A notice period can be considered as a region between trying to supply the market with all the ancient product to satisfy the consumers demands and between starting talking about a new invention that shall come to replace the old product. in other words the big companies plan to supply the market with the max of the products balanced within the consumer demands, from the beginning of the supply to the end, they push the product as much as possible to benefit,now all their products were in the market, the supply is stopped , the push continue to give the stores the chance to buy their products, consumers demands decrease to the maximumk the companies start talking about a new invention to replace the old, than they retired the old product from the market and supply the new one 

Agreed to Sir Raid Shahal.

It is, since it's still a companies profit. Customer Service should have to do something until the supply last and to leave a good product review to the consumers.

Mere customer service may not bring in any difference. I would rather revamp my product by maintaining the existing stakes and investing more on promotions.

yes it worth it

at least if you did not sell, it will leave a passion for your customers that your company is concerned about its customers

on the other hand, pushing the product by giving discount at the end of its cycle will also build loyalty from your customers to your company and to you since you are the one who is offering the discounts 

you may use that product as a gift when customer buy another virgin product 

i don't think this may serve the customer, but may be often a weakness point for this product because of not consuming it at production date