Olympiad problems

1954 Putnam, B2

Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$ Assume as known that this series converges as well and denote its sum by $S$. Denote by $s_k, S_k$ respectively the $k$-th partial sums of both series. Prove that $$ \!\!\!\! \text{i})\; S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$ $$ \text{ii}) \; S\ne s.$$

2006 Cezar Ivănescu, 1

Tags: limit , factorial , harmonic series , newton s binom , integral , calculus

[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^2}\sum_{i=0}^n\sqrt{\binom{n+i}{2}} $ [b]b)[/b] $ \lim_{n\to\infty } \frac{a^{H_n}}{1+n} ,\quad a>0 $

2004 Nicolae Coculescu, 1

Tags: limit , sequence , series , harmonic series , calculus

Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $

1975 Putnam, B6

Tags: inequalities , harmonic series

Let $H_n=\sum_{r=1}^{n} \frac{1}{r}$. Show that $$n-(n-1)n^{-1\slash (n-1)}>H_n>n(n+1)^{1\slash n}-n$$ for $n>2$.