In a medium saucepan, combine the water, sliced ginger, and sugar. Bring the mixture to a simmer over medium heat, stirring until the sugar is completely dissolved. Keep it simmering for 5 minutes to infuse the ginger flavor.

Remove the saucepan from the heat. Add the black tea bags (or loose leaf tea in an infuser) and allow it to steep for 5 to 7 minutes, depending on desired strength. Remove the tea bags/infuser.

While the tea is steeping, place the fresh pineapple chunks into a blender and process until completely smooth. If preferred, strain the pineapple puree through a fine-mesh sieve to remove excess pulp, though keeping the pulp adds body.

Pour the brewed tea mixture into the blender with the pineapple puree. Add the fresh lime juice. Blend briefly for about 10 seconds to combine thoroughly.

Transfer the entire mixture to a pitcher. Taste and adjust sweetness or acidity if necessary. Chill thoroughly in the refrigerator for at least 1 hour, or until completely cold.

To serve, fill tall glasses with ice cubes. Pour the chilled Pineapple Ginger Iced Tea over the ice. Garnish each serving with a fresh pineapple wedge and optionally, a thin slice of ginger.