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Homework Statement
Do the following series converge or diverge?
## \sum_{n=2}^\infty \frac{1}{\sqrt{n} +(-1)^nn}## and
##\sum_{n=2}^\infty \frac{1}{1+(-1)^n\sqrt{n}}##.
Homework Equations
Leibniz convergence criteria:
If ##\{a_n\}_{k=1}^\infty## is positive, decreasing and ##a_n \to 0##, the alternating series ##\sum_{n=1}^\infty (-1)^{n-1}a_n## is convergent.
The Attempt at a Solution
I suspect the first series converges and the second diverges but I need to show that.
Starting with the first series it can be rewritten
##\sum_{n=2}^\infty \frac{1}{\sqrt{n}}\frac{(-1)^n}{(-1)^n+\sqrt{n}}##
At this point I had hopes that there exists an ##N## for which ##\forall n \ge N## ##a_n## is decreasing. That is show that
##\frac{1}{\sqrt{n+1}}\frac{1}{-1+\sqrt{n+1}} \le \frac{1}{\sqrt{n}}\frac{1}{1+\sqrt{n}}##. Sadly based on some numerical experiments this doesn't seem to be true so I need another approach.
Another idea was to be able to pair together the coefficients
##\frac{1}{\sqrt{n+1}}\frac{1}{-1+\sqrt{n+1}}+ \frac{1}{\sqrt{n}}\frac{1}{1+\sqrt{n}}## to show that the partial sum ##\lim_{n\to \infty} S_{2n}## exists but I see no useful way to show this either.