Tag Archives: convergence

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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2014-07 Subsequence

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.

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