Determine Convergence/Divergence of the following series

[Mampac]

Hey, I stumbled upon this series which literally drives me crazy.

Obviously the Root and Ratio Tests are not applicable and lead to incomputable limits. Same goes for the Term Test (gives 0 thus inconclusive).

Remaining tests are Direct Comparison and Limit Comparison tests.

Now by Direct Comparison, i replace the cosine by either 5 or -5, doesn't matter. I get later that the function is similar to 1/n which diverges (factor out root of n, then n out of the sum of n and the root).

However, when I apply the Limit Comparison test, that is, I divide the function by 1/n, I get an undefined limit, which has to be non-zero and positive for them to both converge/diverge.

Gosh this disgusting cosine

 

[lev888]

However, when I apply the Limit Comparison test, that is, I divide the function by 1/n, I get an undefined limit, which has to be non-zero and positive for them to both converge/diverge.

Why is this a problem? The fact that the limit is undefined does not tell us anything. If the other test indicates that the series diverges, then it diverges.

 

[lex]

Yes. That simply means that this particular test doesn't help in this particular case. You only need one test to work. You can show that it diverges, by comparison with [MATH]\tfrac{1}{n}[/MATH]
[MATH]n+\sqrt{n-1} ≤ 2n[/MATH][MATH]\therefore \frac{1}{n+\sqrt{n-1}}≥\frac{1}{2n}[/MATH]
[MATH]7+5\cos{n^2}≥2[/MATH]
[MATH]\therefore \frac{7+5\cos{n^2}}{n+\sqrt{n-1}}≥\frac{1}{n}[/MATH]
which diverges