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Given the graph below, does

converge or diverge?

We know

converges, which means the region between the graph of and the x-axis on [1,∞) has finite area. Call this region B.

Since the region between the graph of f ( x ) and the x-axis on [1,∞) is contained within region B, its area must also be finite. This means

converges.

diverges, which means the region under on the interval (0,1] has infinite area. Call this region A.

Since the region under f ( x ) on that interval is even larger than region A, its area must be infinite also, which means

must diverge.

diverges, which means the area under on the interval (0,1] is infinite.

Unfortunately, that doesn't tell us anything about whether the area under f ( x ) on that interval is finite or infinite.

There's a property for definite integrals that says if f ( x ) < g ( x ) for all x in [a,b], then