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\[ \displaystyle \sum_{n=1}^{\infty} \dfrac{n^23^n}{4^n} = ? \]

\[ \displaystyle \sum_{n=0}^{\infty} \dfrac{(n!)^k}{(nk)!} x^n \]

For positive integer values of \( k \), what is the radius of convergence of the function above?

\[ \large \sum_{n=2}^{\infty} \dfrac{x^{2n}}{n ( \ln n )^2 } \]

Find the interval of convergence of the infinite sum above.

\[\large \begin{align} A & = \sum_{n=1}^{\infty} \dfrac{(-1)^n \cdot n}{\sqrt{n^3+4}} \\ B & = \sum_{n=1}^{\infty} \dfrac{(2n)!}{(n!)^2} \\ C & = \sum_{n=0}^{\infty} \dfrac{1 + \sin(n^e)}{e^n} \end{align} \]

Let \( z \in \mathbb{R} \) be the smallest positive value that the function \( y = y(x) = x^x \) can attain, for \( x > 0 \).

What is the ...

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