Here is a specific diverging alternating series I came across: $$\sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n}+(-1)^n}$$ This series diverges, one can show that: $$\frac{(-1)^n}{\sqrt{n}+(-1)^n}=\frac{(-1)^n}{\sqrt{n}}+\frac{1}{n}+O(\frac{1}{n\sqrt{n}})$$ and the first term is a converging alternating series, the third term is a Riemann converging series, and so the middle term blows up to infinity, making the overall series blow up to infinity.
I understand perfectly well the computation part, but I am very disturbed by the way I get the wrong intuition from the general term. If I look at two successive terms of the series, I get the overall feeling that the even terms are always smaller in absolute values than the uneven terms. And so my guess, before jumping into computation, would have been to say that should the series diverge, it should blow up to minus infinity.
Quite clearly, my guess would have been wrong. Why is my gut betraying me so badly ? I must have made a really bad mistake in my estimation.
Thank you very much.
Correction and answer:
In comments geetha290krm pointed out the formula in my question is incorrect, it should read: $$\frac{(-1)^n}{\sqrt{n}+(-1)^n}=\frac{(-1)^n}{\sqrt{n}}-\frac{1}{n}+O(\frac{1}{n\sqrt{n}})$$
I was reading a maths textbook for undergraduates, and, I blindly trusted the book because I understood how the formula was derived but did not verify the sign and correctness of the final formula. In the end, I was detecting a mistake in the textbook, but was refusing to question the textbook. My gut feeling was correct.
Thank you very much for pointing out the mistake !
