Does
converge or diverge? If it converges, what does it converge to?

Answer

Look at the sequence of integrals

as b gets larger:

These values are getting closer and closer to 1.

As *b* approaches ∞,

approaches 1. This means

converges to 1.

Each integral in the above exercise is the area between the graph of and the *x*-axis on an interval with left endpoint 1.

As *b* approaches ∞, the area

approaches the total area between the graph of

and the *x*-axis on the interval [1,∞).

In symbols, the total area between and the *x*-axis on [1,∞) is

We abbreviate this limit by writing