Question

Suppose a computer can sum one million terms per second of the divergent series $\sum _{n=1}^N\frac{1}{n}.$ Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed 100 .

Short answer

It will take approximately $e^{100}/{10}^6$ seconds.

Step-by-step solution

Understanding the Problem

We are given a divergent series $\sum _{n=1}^N\frac{1}{n}$ and need to approximate how long it will take for the partial sum to exceed 100 using the integral test. This test helps approximate the behavior of series by comparing it with an integral.

Setting up the Integral Test

The integral test involves comparing the series $\sum _{n=1}^N\frac{1}{n}$ with the integral ${\int }_1^N\frac{1}{x}dx$. We need to find $N$ such that the sum exceeds 100.

Calculating the Integral

Compute the integral ${\int }_1^N\frac{1}{x}dx$, which gives $\ln (N)$. This integral helps determine the value of $N$ needed to reach the desired partial sum approximation.

Solving for N

Set the integral result equal to 100: $\ln (N)=100$. Solve for $N$ by exponentiating both sides. Thus, $N=e^{100}$.

Estimating Time

Each sum term $\sum _{n=1}^N$ takes one second per million terms. So, the time $T$ to evaluate up to $N=e^{100}$ is $T=\frac{N}{{10}^6}$ seconds.

Key concepts

Divergent Series

A divergent series is a sequence of numbers whose sum grows without bound, meaning it does not converge to a particular finite value. This can be confusing because, at first glance, some divergent series seem like they might have a limit due to the decreasing nature of their terms. However, for a series to be divergent, the sum of its terms increases indefinitely as more terms are added.

In mathematical terms, a series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ is divergent if the sequence of partial sums $S_N=\sum _{n=1}^N{a}_n$ does not converge to a limit as $N\to \mathrm{\infty }$. This concept is essential when dealing with infinite series, as it helps us understand whether or not a series can be summed to a defined value.

Knowing whether a series is divergent can help mathematicians figure out useful approximations and behaviors of the series, such as determining whether a practical computation is feasible within specific parameters, like time constraints.

Harmonic Series

The harmonic series is one example of a divergent series. It is represented as $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$. Every term in the harmonic series decreases as $n$ increases, yet intriguingly, this series does not converge, and its sum grows without bound.

The name "harmonic" comes from its relation to harmonic functions in music and acoustics. Despite its divergence, the harmonic series is valuable in mathematics due to applications in various fields, such as statistics and number theory.

• The terms$\frac{1}{n}$ approach zero as $n\to \mathrm{\infty }$.
• Despite the small size of terms, the series sums grow indefinitely.
• This series is critical in understanding real-world phenomena where initial rapid growth slows down over time but never fully stabilizes.

Being familiar with the harmonic series provides a solid foundation for exploring more complex mathematical concepts and understanding the properties of divergent series further.

Approximation with Integrals

The Integral Test is a powerful method for approximating the sum of series, especially when determining the behavior of a series that is not straightforward. By using integration, we can estimate how many terms of the series need to be added to approximate a certain value or limit.

The core idea of this test is comparing the sum $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ with the integral ${\int }_1^{N}f(x)dx$ where $f(n)=a_n$. This approach makes use of continuous calculus to gain insights into discrete problems, such as the harmonic series.

To approximate the series $\sum _{n=1}^N\frac{1}{n}$, we can use the integral ${\int }_1^N\frac{1}{x}dx=\ln (N)$. This calculation gives an easy way to find a rough estimate of any number of series terms needed for a desired sum and is extremely useful for practical calculations.

This method allows you to understand complex series behaviors without having to compute each term individually, streamlining the process significantly. It is particularly helpful when dealing with series like the harmonic series, where sums can become incredibly large.