MathDB

i) Prove that for every sequence

(a_{n})_{n\in \mathbb{N}}

, such that

a_{n}>0

for all

n \in \mathbb{N}

and

\sum_{n=1}^{\infty}a_{n}<\infty

, we have

\sum_{n=1}^{\infty}(a_{1}a_{2} \cdots a_{n})^{\frac{1}{n}}< e\sum_{n=1}^{\infty}a_{n}.

ii) Prove that for every

\epsilon>0

there exists a sequence

(b_{n})_{n\in \mathbb{N}}

such that

b_{n}>0

for all

n \in \mathbb{N}

and

\sum_{n=1}^{\infty}b_{n}<\infty

and

\sum_{n=1}^{\infty}(b_{1}b_{2} \cdots b_{n})^{\frac{1}{n}}> (e-\epsilon)\sum_{n=1}^{\infty}b_{n}.

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