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Let \(\displaystyle f(x) = \sum_{n=0}^{\infty} x^{n^k}\), where \(k \in \mathbb{Z}^+\)

Prove That ...

Prove That

\[\sum_{r=1}^{n} \dbinom{n}{r} H_{r} = 2^{n}\left( H_{n} - \sum_{r=1}^{n} \dfrac{1}{r 2^r} \right) \]

\[\large f(x) = \sum_{n=0}^{\infty} x^{n^2}, \quad \lim_{x \to 1^-} \sqrt{1-x} f(x) = \frac{\pi^a}{b} \]

The above holds for rational numbers ...

Let \(f : \mathbb{N} \to \mathbb{N}\) be a strictly increasing function such that \(f(2) = 8\) and \(f(ab) = f(a) \cdot f(b)\) for ...

Let \(f : \mathbb{N} \to \mathbb{N}\) be a strictly increasing function such that \(f(2) = 2\) and \(f(ab) = f(a) \cdot f(b)\) for ...

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