Is this too simple?
Put a volt meter across the battery with the engine not running. Take a reading.
Start the engine and take another reading.
If the second reading is the same as or lower than the first, you can be 99% sure that the alternator is not charging.
The easist way to be sure that it isn't charging? Disconnect the positives while the engine is running. If it was charging before, it isn't any more.
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"Two possibilities exist: Either we are alone in the Universe or we are not. Both are equally terrifying."
Arthur C. Clarke