Problem 11 - Single Variable Calculus

Question

Determine whether the \(p\) -series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1.01}} $$

Step-by-Step Solution

Answer

The given \(p\)-series is \(\sum_{n=1}^{\infty} \frac{1}{n^{1.01}}\) with \(p = 1.01\). Since \(p > 1\), the series converges.

Step 1: Identifying the value of p

In the given \(p\)-series, we have: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1.01}} $$ Comparing this with the general form of a \(p\)-series, we can see that the value of \(p\) is \(1.01\).

Step 2: Comparing p with 1

Now, let us compare the value of \(p\) with \(1\): $$ p = 1.01 > 1 $$

Step 3: Determining convergence or divergence

Since \(p > 1\), according to the \(p\)-series test for convergence, the given series converges. So, the \(p\)-series $$ \sum_{n=1}^{\infty} \frac{1}{n^{1.01}} $$ is convergent.

Key Concepts

Series Convergence

In mathematics, understanding whether a series converges or diverges is crucial. A series is essentially the sum of the terms of a sequence.
For a series to converge, the sum of its terms must approach a fixed number as more terms are added. Conversely, a divergent series grows without bound, never settling at a particular value.
Knowing if a series converges or diverges helps in many fields such as physics, computer science, and economics.

• A convergent series means predictable behavior.
• Divergent series indicate potential for infinite growth.

In our exercise, determining convergence means checking if the use of the infinity symbol indicates that all added terms tend toward a specific sum.

p-series Test

The p-series test is a straightforward tool to determine the convergence of specific infinite series. A p-series is a particular type of series that takes the form: \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] Here, \(p\) is a constant. The p-series test is based on a simple rule: a p-series converges if \(p > 1\), and it diverges if \(p \leq 1\).
This is because the terms of the series become small enough quickly to add up to a finite number when \(p\) is larger than 1.

• If \(p = 2\), or any number greater than 1, the series converges.
• If \(p = 1\), the series is known as the harmonic series, which diverges.

So in our case, with \(p = 1.01\), the series converges since \(1.01 > 1\). This clear guideline helps mathematicians quickly assess the nature of p-series.

Calculus

Calculus is a branch of mathematics that focuses on how things change. It's crucial for studying motion, growth, and change in various scientific fields.
With series, calculus provides the tools and theories to tackle concepts of convergence and divergence. These are fundamental in understanding continuous growth and cumulative effects.
Some branches of calculus that often deal with series include:

• Differential calculus, which studies rates of change.
• Integral calculus, which focuses on accumulation of quantities.

Using calculus, mathematicians can solve complex real-world problems, model natural phenomena, and even hone financial predictions. Understanding series like the p-series is a stepping stone in calculus to achieve these deeper insights.

Infinite Series

An infinite series is an endless summation of terms from a sequence. It builds upon the idea of adding numbers without limit.
In practice, an infinite series can be written as: \[ \sum_{n=1}^{\infty} a_n \] where \(a_n\) represents each term of the series.
Infinite series can be mind-bending because they suggest you add values into infinity. Convergence and divergence describe if this endless addition settles to a finite number or continues to grow.

• Convergent series find a stable sum.
• Divergent series don't stabilize or reach a limit.

The p-series we see in the example is an infinite series. Using tests like the p-series test, mathematicians can determine if the sum ends up more as a predictable or ever-expanding quantity.