"Jesus Christ who was of the race of David, who was the Son of Mary, who was truly born and ate and drank, was truly persecuted under Pontius Pilate, was truly crucified and died in the sight of those in heaven and on earth and those under the earth; who moreover was truly raised from the dead, His Father having raised Him, who in the like fashion will so raise us also who believe on Him." - Ignatius, (written around AD 110-115.)
The early disciples had a life changing encounter with Jesus. As I have studied the life of Jesus, I have found that many ancient non-Christian historians believe Jesus' early disciples were convinced of his resurrection from the dead. And the immediate and rapid growth of Christianity is hard to explain apart from Jesus' resurrection. I've found ancient non-biblical accounts of the earthquake and darkness that covered the land at the time of the crucifixion. Many tried to explain the events of this time from a natural perspective, but they were clearly supernatural. I, like many of you have had a spiritual encounter with the resurrected Christ, just as the early Church Fathers that I quoted above had. These men did not give up their lives to follow a lunatic.
I believe in the resurrection of Jesus because of my personal life changing experience, and the confidence I have in the authority of Scripture. The gospels are full of details about Jesus and eyewitness accounts and were written as history, and they were written no more than forty to sixty years after Jesus' death. Paul wrote his letters about fifteen to twenty-five years after Jesus' resurrection, and they confirm the gospel accounts of Jesus. The Jews of the first century were well trained in the art of memorization and the disciples of teachers during that time committed their teaching to memory. They communicated events, that took place verbally. As most of the eye witnesses were dying off, there was a need to write these accounts down for future generations. The death of Jesus was the most recorded event, not only by Christians but also by non-Christians in ancient history.
Here is one example that Tim Keller gives of details that point to the authenticity of the eyewitness accounts, "Mark for example, says that the man who helped Jesus carry his cross to Calvary "was the father of Alexander and Rufus" (Mark 15:21). There is no reason for the author to include such names unless the readers know or could have access to them. Mark is also saying, "Alexander and Rufus vouch for the truth of what I am telling you, if you want to ask them." - The Reason For God, page 101. - Bobby
Sources: The Reason For God, Tim Keller
The Historical Jesus, Gary Habermas