This phase III trial studies magnetic resonance imaging (MRI) surveillance and prophylactic cranial irradiation (PCI) to see how well they work compared to MRI surveillance alone in treating patients with small cell lung cancer. MRI scans are used to monitor the possible spread of the cancer with an MRI machine over time. PCI is radiation therapy that is delivered to the brain in hopes of preventing spread of cancer into the brain. The use of brain MRI alone may reduce side effects of receiving PCI and prolong patients' lifespan. Monitoring with MRI scans alone (delaying radiation until the actual spread of the cancer) may be at least as good as the combination of PCI with MRI scans.