Olympiad problems

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1953 Putnam, B1

Tags: series , convergence

Is the infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ convergent?

ICMC 6, 6

Tags: real analysis , convergence , floor function , sequence

Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges. [i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point. [i]Proposed by Ethan Tan[/i]

ICMC 7, 4

Tags: convergence , series , real analysis

Let $(t_n)_{n\geqslant 1}$ be the sequence defined by $t_1=1, t_{2k}=-t_k$ and $t_{2k+1}=t_{k+1}$ for all $k\geqslant 1.$ Consider the series \[\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.\]Prove that this series converges to a positive real number. [i]Proposed by Dylan Toh[/i]

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