Ratio test for $\frac{\sqrt{n^n}}{2^n}$ February 11    2

Before I tried root test, I did ratio test for $\frac{\sqrt{n^n}}{2^n}$ and got: $$\lim_{n\to\infty}\frac{\sqrt{(n+1)^{(n+1)}}}{2^{n+1}}\cdot \frac{2^n}{\sqrt{n^n}} = \lim_{n\to\infty} \frac{1}{2}\cdot \sqrt{\frac{(n+1)^{(n+1)}}{n^n}} = \frac{1}{2}$$

Ratio test: $n^\sqrt{n}$ February 11    4

I need to determine the radius of convergence of: $$\sum_{n=1}^\infty z^n n^\sqrt{n}$$I have, by use of the ratio test, written: (Because I know it tends to 1) $$\lim_{n\to\infty} \frac{n^\sqrt{n}}{(n+1)^\sqrt{n+1}}\to 1$$But I am unable to demonstra

Domain of \lim_{n\to\infty}(n+1)x 1 February 11

In the process of doing the ratio test (for testing convergence of a function), I have the following issue. I am trying to find the convergence domain of $x$ in the following function: $$ \begin{align} \lim_{n\to\infty}|(n+1)x| < 1 \end{align} $$The

Inference Rules, Not(P) Implies Not(Q) / Q Implies P February 11    1

Inference Rules, Not(P) Implies Not(Q) / Q Implies P
I do not understand Implication and Inference, I am going over the MIT Computer Science course and they have this part in their lecture notes, why is the second rule not a logical deduction? ...

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