17: Using the fact that "exponentials dominate powers" to handle the difference of infinities in the logarithm, it is easy to see that the expression is of the type ∞ − ∞:

We also saw that the logarithm term in fact behaves like 2n when n grows large, so eventually the term "-n²" should prevail; that is, the guess should be that the sequence converges to negative infinity.

How to prove it? Here factoring out the dominant power should help, but perhaps an easier idea is to put the two terms together. This is not a difference of roots, so that trick is out of the question. One could try the universal trick for changing difference into ratio (see the box "indeterminate difference"), but that would be beastly. Far better choice is to try to change the difference into one expression using some algebra. One trick seems quite obvious; try it.

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