docs: migrate testing tutorial from on_event to lifespan

docs/en/docs/advanced/async-tests.md

@@ -94,6 +94,6 @@ As the testing function is now asynchronous, you can now also call (and `await`)
 
 /// tip
 
-If you encounter a `RuntimeError: Task attached to a different loop` when integrating asynchronous function calls in your tests (e.g. when using <a href="https://stackoverflow.com/questions/41584243/runtimeerror-task-attached-to-a-different-loop" class="external-link" target="_blank">MongoDB's MotorClient</a>), remember to instantiate objects that need an event loop only within async functions, e.g. an `@app.on_event("startup")` callback.
+If you encounter a `RuntimeError: Task attached to a different loop` when integrating asynchronous function calls in your tests (e.g. when using <a href="https://stackoverflow.com/questions/41584243/runtimeerror-task-attached-to-a-different-loop" class="external-link" target="_blank">MongoDB's MotorClient</a>), remember to instantiate objects that need an event loop only within async functions, for example, using a lifespan async context manager.
 
 ///

docs_src/app_testing/tutorial003_py310.py

@@ -1,15 +1,19 @@
+from contextlib import asynccontextmanager
+
 from fastapi import FastAPI
 from fastapi.testclient import TestClient
 
-app = FastAPI()
-
 items = {}
 
 
-@app.on_event("startup")
-async def startup_event():
+@asynccontextmanager
+async def lifespan(app: FastAPI):
     items["foo"] = {"name": "Fighters"}
     items["bar"] = {"name": "Tenders"}
+    yield
+
+
+app = FastAPI(lifespan=lifespan)
 
 
 @app.get("/items/{item_id}")