feat(examples): multi-server chatbot with tool routing (closes #740)

[Nishant-Chaudhary5338]

Summary

Adds a runnable multi-server chatbot example showing how one MCP client connects to multiple servers simultaneously and routes Claude's tool calls to the correct server.

Closes #740

What this adds

• examples/server/src/weatherServer.ts — stateless Streamable HTTP server on :3001 with get_weather and get_forecast tools
• examples/server/src/mathServer.ts — stateless Streamable HTTP server on :3002 with add, multiply, and convert_temperature tools
• examples/client/src/multiServerChatbot.ts — connects to both servers, builds a Map<toolName, Client> routing table, runs an interactive Anthropic SDK agentic chat loop, and dispatches parallel tool calls to the correct server

How it works

Client ──► Map<toolName, Client> ──► weather-server (:3001)
                                  └► math-server   (:3002)

1. On startup the client calls listTools() on each server and populates the routing table
2. For every Anthropic tool_use block, the client looks up the owning server in the map and dispatches the call — no hardcoded routing, no tool-name prefixes needed
3. Multiple tool calls in a single response are dispatched in parallel via Promise.all

How to run

Terminal 1:

pnpm --filter @modelcontextprotocol/examples-server exec tsx src/weatherServer.ts

Terminal 2:

pnpm --filter @modelcontextprotocol/examples-server exec tsx src/mathServer.ts

Terminal 3:

ANTHROPIC_API_KEY=sk-... \
  pnpm --filter @modelcontextprotocol/examples-client exec tsx src/multiServerChatbot.ts

Try these prompts to exercise both servers in one turn:

What's the weather in Tokyo?
What is 17 x 19?
Convert 100 C to Fahrenheit and give me a 3-day forecast for Paris.

Testing

• pnpm --filter @modelcontextprotocol/examples-client run check (typecheck + lint + prettier) passed
• pnpm --filter @modelcontextprotocol/examples-server run check passed
• All 5 tools verified via direct JSON-RPC calls
• All 6 temperature conversion paths correct
• Boundary values (days=1, days=7), negative numbers, floats, multiply-by-zero
• Validation errors (days>7, invalid enum) return isError: true
• Client connection failure prints a clear actionable error message

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