# 2020-04 Convergence at all but one point

Let $$f_n : [-1, 1] \to \mathbb{R}$$ be a continuous function for $$n = 1, 2, 3, \dots$$. Define
$g_n(y) := \log \int_{-1}^1 e^{y f_n(x)} dx.$
Suppose there exists a continuous function $$g: \mathbb{R} \to \mathbb{R}$$ and $$y_0 \in \mathbb{R}$$ such that $$\lim_{n \to \infty} g_n(y) = g(y)$$ for all $$y \neq y_0$$. Prove or disprove that $$\lim_{n \to \infty} g_n(y_0) = g(y_0)$$.

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# 2019-05 Convergence with primes

Let $$p_n$$ be the $$n$$-th prime number, $$p_1 = 2, p_2 = 3, p_3 = 5, \dots$$. Prove that the following series converges:
$\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.$

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Let $$a_1,a_2,\ldots$$ be an infinite sequence of positive real numbers such that $$\sum_{n=1}^\infty a_n$$ converges. Prove that for every positive constant $$c$$, there exists an infinite sequence $$i_1<i_2<i_3<\cdots$$ of positive integers such that $$| i_n-cn^3| =O(n^2)$$ and  $$\sum_{n=1}^\infty \left( a_{i_n} (a_1^{1/3}+a_2^{1/3}+\cdots+a_{i_n}^{1/3})\right)$$ converges.