Find the first five partial sums of the given series and...

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=3}^{\infty} \frac{\ln n}{e^{n}}$$

Key Concepts

Relative Growth Rates of Functions

Understanding how different functions grow is crucial when analyzing series. Exponential functions grow or decay much faster than logarithmic functions, which means that in series where a logarithmic term is divided by an exponentially decaying term, the rapid decay can dominate the slow growth of the logarithmic term and promote convergence.

Infinite Series and Partial Sums

An infinite series is the sum of the terms of an infinite sequence. To study the convergence of a series, mathematicians look at its partial sums—the sum of the first n terms—which reveal whether the series approaches a specific value as n increases. Calculating partial sums is fundamental to understanding the behavior of series and assessing their convergence or divergence.

Convergence and Divergence of Series

Convergence refers to an infinite series approaching a finite limit as more terms are added, while divergence means that the series does not settle to a finite value. Determining convergence involves analyzing whether the sequence of partial sums stabilizes. Various tests, such as the Ratio Test and the Comparison Test, are used to assess the convergence properties of a series.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value, leading to a rapid decrease as the variable increases. In series, when the denominator exhibits exponential behavior, the terms of the series tend to shrink quickly, which is a strong indicator that the series might converge.

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