The foundation and emphasis of robotic manipulator control is Inverse Kinematics (IK). Due to the complexity of derivation, difficulty of computation, and redundancy, traditional IK solutions pose numerous challenges to the operation of a variety of robotic manipulators. This paper develops a Deep Reinforcement Learning (RL) approach for solving the IK problem of a 7-Degree of Freedom (DOF) robotic manipulator using Product of Exponentials (PoE) as a Forward Kinematics (FK) computation tool and the Deep Q-Network (DQN) as an IK solver. The selected approach is architecturally simpler, making it faster and easier to implement, as well as more stable, because it is less sensitive to hyperparameters than continuous action spaces algorithms. The algorithm is designed to produce joint-space trajectories from a given end-effector trajectory. Different network architectures were explored and the output of the DQN was implemented experimentally on a Sawyer robotic arm. The DQN was able to find different trajectories corresponding to a specified Cartesian path of the end-effector. The network agent was able to learn random Bézier and straight-line end-effector trajectories in a reasonable time frame with good accuracy, demonstrating that even though DQN is mainly used in discrete solution spaces, it could be applied to generate joint space trajectories.