docs: update README with lifespan examples and usage

Add comprehensive documentation for lifespan support:
- Add usage examples for both Server and FastMPC classes
- Document startup/shutdown patterns
- Show context access in tools and handlers
- Clean up spacing in test files

README.md

@@ -135,6 +135,28 @@ mcp = FastMCP("My App")
 
 # Specify dependencies for deployment and development
 mcp = FastMCP("My App", dependencies=["pandas", "numpy"])
+
+# Add lifespan support for startup/shutdown
+@asynccontextmanager
+async def app_lifespan(server: FastMCP) -> AsyncIterator[dict]:
+    """Manage application lifecycle"""
+    try:
+        # Initialize on startup
+        await db.connect()
+        yield {"db": db}
+    finally:
+        # Cleanup on shutdown
+        await db.disconnect()
+
+# Pass lifespan to server
+mcp = FastMCP("My App", lifespan=app_lifespan)
+
+# Access lifespan context in tools
+@mcp.tool()
+def query_db(ctx: Context) -> str:
+    """Tool that uses initialized resources"""
+    db = ctx.request_context.lifespan_context["db"]
+    return db.query()
 ```
 
 ### Resources
@@ -334,7 +356,39 @@ def query_data(sql: str) -> str:
 
 ### Low-Level Server
 
-For more control, you can use the low-level server implementation directly. This gives you full access to the protocol and allows you to customize every aspect of your server:
+For more control, you can use the low-level server implementation directly. This gives you full access to the protocol and allows you to customize every aspect of your server, including lifecycle management through the lifespan API:
+
+```python
+from contextlib import asynccontextmanager
+from typing import AsyncIterator
+
+@asynccontextmanager
+async def server_lifespan(server: Server) -> AsyncIterator[dict]:
+    """Manage server startup and shutdown lifecycle."""
+    try:
+        # Initialize resources on startup
+        await db.connect()
+        yield {"db": db}
+    finally:
+        # Clean up on shutdown
+        await db.disconnect()
+
+# Pass lifespan to server
+server = Server("example-server", lifespan=server_lifespan)
+
+# Access lifespan context in handlers
+@server.call_tool()
+async def query_db(name: str, arguments: dict) -> list:
+    ctx = server.request_context
+    db = ctx.lifespan_context["db"]
+    return await db.query(arguments["query"])
+```
+
+The lifespan API provides:
+- A way to initialize resources when the server starts and clean them up when it stops
+- Access to initialized resources through the request context in handlers
+- Support for both low-level Server and FastMCP classes
+- Type-safe context passing between lifespan and request handlers
 
 ```python
 from mcp.server.lowlevel import Server, NotificationOptions

tests/server/fastmcp/test_func_metadata.py

@@ -236,7 +236,7 @@ async def test_lambda_function():
 
 def test_complex_function_json_schema():
     """Test JSON schema generation for complex function arguments.
-    
+
     Note: Different versions of pydantic output slightly different
     JSON Schema formats for model fields with defaults. The format changed in 2.9.0:
 
@@ -245,16 +245,16 @@ def test_complex_function_json_schema():
          "allOf": [{"$ref": "#/$defs/Model"}],
          "default": {}
        }
-       
+
     2. Since 2.9.0:
        {
          "$ref": "#/$defs/Model",
          "default": {}
        }
-       
+
     Both formats are valid and functionally equivalent. This test accepts either format
     to ensure compatibility across our supported pydantic versions.
-    
+
     This change in format does not affect runtime behavior since:
     1. Both schemas validate the same way
     2. The actual model classes and validation logic are unchanged
@@ -262,17 +262,17 @@ def test_complex_function_json_schema():
     """
     meta = func_metadata(complex_arguments_fn)
     actual_schema = meta.arg_model.model_json_schema()
-    
+
     # Create a copy of the actual schema to normalize
     normalized_schema = actual_schema.copy()
-    
+
     # Normalize the my_model_a_with_default field to handle both pydantic formats
-    if 'allOf' in actual_schema['properties']['my_model_a_with_default']:
-        normalized_schema['properties']['my_model_a_with_default'] = {
-            '$ref': '#/$defs/SomeInputModelA',
-            'default': {}
+    if "allOf" in actual_schema["properties"]["my_model_a_with_default"]:
+        normalized_schema["properties"]["my_model_a_with_default"] = {
+            "$ref": "#/$defs/SomeInputModelA",
+            "default": {},
         }
-    
+
     assert normalized_schema == {
         "$defs": {
             "InnerModel": {