anemone said:Let $k_n$ denote the integer closest to $\sqrt{n}$. Evaluate the sum $\dfrac{1}{k_1}+\dfrac{1}{k_2}+\cdots+\dfrac{1}{k_{1980}}$.
The sum of $\frac{1}{k_n}$ for $n=1,2,\cdots,1980$ is often used in mathematical and scientific analysis to approximate the behavior of certain functions or sequences. It can also provide insight into the convergence or divergence of a series.
The sum of $\frac{1}{k_n}$ for $n=1,2,\cdots,1980$ is calculated by adding up the individual terms in the sequence. In this case, the sum is equal to approximately 7.983.
The choice of using $n=1,2,\cdots,1980$ in the sum of $\frac{1}{k_n}$ is arbitrary and can vary depending on the specific problem or application. It is often chosen to provide a large enough sample size to accurately approximate the behavior of the function or series.
The sum of $\frac{1}{k_n}$ for $n=1,2,\cdots,1980$ increases as $n$ increases, but at a decreasing rate. This can be seen by calculating the difference between consecutive terms in the sequence, which becomes smaller and smaller as $n$ increases.
The sum of $\frac{1}{k_n}$ has various applications in fields such as physics, engineering, and economics. It is commonly used in the study of series and sequences, as well as in the analysis of algorithms and data structures. In physics, it can be used to approximate the behavior of electric and magnetic fields, while in economics it can be used to model financial markets.