This phase II trial studies how well pembrolizumab after standard treatment with radiation plus the following chemotherapy drugs: cisplatin or carboplatin, plus etoposide works in treating patients with limited stage small cell lung cancer (LS-SCLC). Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving pembrolizumab after standard treatment with radiation plus chemotherapy may increase the ability of the immune system to fight LS-SCLC.