Problem 48 Hard Difficulty

(a) If $ g(x) = \frac{1}{(\sqrt{x} - 1)} $, use your calculator or computer to make a table of approximate values of $ \displaystyle \int_2^t g(x)\ dx $ for $ t = 5, 10, 100, 1000 $, and $ 10,000 $. Does it appear that $ \displaystyle \int_2^\infty g(x)\ dx $ is convergent or divergent?
(b) Use the Comparison Theorem with $ f(x) = \frac{1}{\sqrt{x}} $ to show that $ \displaystyle \int_2^\infty g(x)\ dx $ is divergent.
(c) Illustrate part (b) by graphing $ f $ and $ g $ on the same screen for $ 2 \le x \le 20 $. Use your graph to explain intuitively why $ \displaystyle \int_2^\infty g(x)\ dx $ is divergent.

Answer

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Video Transcript

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