What Test?

Calculus Level 3

Let f ( x ) = x − 4 x − 19 f( x) =\sqrt{x-4\sqrt{x-19}} and f ′ ( x ) = a x f'( x) =a_{x} . Consider the series A = ∑ n = 20 ∞ a n A=\displaystyle \sum_{n=20}^\infty {a_{n}} . Determine if A A converges or diverges.

Converges Diverges

David Holcer by David Holcer , 20, Canada

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2 solutions

Chris Lewis
May 20, 2019

Answer to the problem's title: the integral test

Since the sequence is defined in terms of a derivative, this is easy to use; f ( x ) → ∞ f(x) \to \infty as x → ∞ x \to \infty so the series diverges .

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Mark Hennings
May 19, 2019

Since f ′ ( x ) = 1 2 x − 4 x − 19 [ 1 − 2 x − 19 ] f'(x)\; =\; \frac{1}{2\sqrt{x- 4\sqrt{x-19}}}\left[1 - \frac{2}{\sqrt{x-19}}\right] we see that lim ⁡ x → ∞ x f ′ ( x ) = 1 2 \lim_{x \to \infty}\sqrt{x}f'(x)\; = \; \tfrac12 and so, for large enough x > 0 x > 0 , x f ′ ( x ) > 1 4 \sqrt{x}f'(x)>\tfrac14 , so that f ′ ( x ) > 1 4 x f'(x)>\tfrac{1}{4\sqrt{x}} . Thus A A diverges.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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