Is Rama really perfect?

In reading the second part of the Ramayana, I was surprised when I came to the part about Rama questioning the purity of Sita. The reason this was so surprising was that it showed a weakness on Rama's part. This weakness is trust. He does not trust Sita when she tells him that she has been pure while in captivity. It requires a rigorous test such as walking into fire in front of a great audience and also the gods before he will believe that she is pure. However it is somewhat unclear as to whether Rama really does not believe Sita or whether he is influenced by the watchful eyes of the public. It could be that he is so worried about how his marriage and the purity of his wife appears to the public, that he would rather please the crowd than accept the consequences of believing his wife. What do you think? Is Rama's one weakness his trouble believing others or is it his willingness to give into the beliefs of those around him?

  Either way, it is definitely a character flaw and a fault that conflicts with dharma. If we had not just learned that Rama himself is a incarnation of the god Vishnu then it would be a little easier to accept this flaw, but in this case it causes the reader to seriously question everything that Rama and dharma stand for. Can the gods choose to not follow dharma...? Or is it a misinterpretation of the story that is leading to the misconception that Rama has failed to follow dharma in not believing Sita? There are alot of loose ends in the story, especially considering after revealing this flaw, the story goes on to say that Rama lived  for thousands of years and the kingdom was very productive and wealthy. This would seem to indicate that although Rama failed his wife in a very crucial situation, he still all kinds of benefits as king. There seem to be no repercussions of this fault.