Unfortunately we do not have testing software for the processor.
The easiest way to determine if the processor is the defective component is by testing it on another known working and compatible motherboard to see if it causes the same behavior.
For future reference keep in mind that Intel will not provide warranty service for products that have been re-sold through a private party or an online auction site such as www.ebay.com, or similar. This is because we cannot know if the product was functioning when it was sold, or how it was treated/used prior to being sold. In this case, your claim is with the person selling the product or the site you purchased the product through.