Answered prayer may seem like the least rational reason to believe in God because it is difficult to prove. Is it just a coincidence when we petition God for something and our request is granted? How many times must one experience these coincidences before we understand there is real power in prayer. This has been my experience as well as the experience of many others I know.
Where many struggle is when we it appears God is deaf to our prayers. One problem is we often have our relationship with God backwards. In other words if God gives us what we want then He may deserve our attention. I ask anyone who has raised a child how far would your child get if he or she expected you to give them everything they want, no matter how harmful or foolish their request may be. How far would they get if they would only be responsive to you if you gave them what they wanted?
There are countless scriptures explaining God cares for those who follow and love Him. Just as any child who shows gratefulness and love toward their parents, their parents will do everything they can to give that child a good and fruitful life. This does not mean God will ignore the pleas of those hurting so bad they reach out to God just in case there is one. God does want to draw us to Him! We just need to be careful our heart is in the right place when we reach out to God.
There are many books on prayer. I have provided links for two - one book is about how to pray and the other deals with what may seem like unanswered prayers.
How to Pray: A Simple Guide for Normal People by Pete Greig
God on Mute: Engaging the Silence of Unanswered Prayer by Pete Greig