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Problem 1.Let $(a_n)$ and $(b_n)$ be sequences of non-negative real numbers which are not identically zero. Also, let $(c_n)$ be the Cauchy product of these sequences:

$$ c_n = (a \ast b)_{n} = \sum_{k=0}^{n} a_k b_{n-k}. $$

Prove that

$$ \limsup_{n\to\infty} c_{n}^{1/n} = \max \left\{ \limsup_{n\to\infty} a_{n}^{1/n}, \limsup_{n\to\infty} b_{n}^{1/n} \right\}. $$

Problem 2. Let $f : \Bbb{R} \to \Bbb{R}$ be a function such that

$\displaystyle \sum f(a_n)$ converges whenever $\displaystyle \sum a_n$ converges.

Prove that $f$ is linear near $x = 0$.

Proof. Refer to the following attachment:

doc_019.pdf

Problem 3. Solve the functional equation

$$ f(f(x)) - 2f(x) + x = 0, \quad \forall x \in \Bbb{R}$$

where $f : \Bbb{R} \to \Bbb{R}$ is continuous.

Problem 4. Find the limit

$$ \lim_{n\to\infty} \frac{1}{\log n} \sum_{j=1}^{n} \sum_{k=1}^{n} \frac{j+k}{j^3 + k^3}.$$