Question

A computer program gives the result \(1 / 6\) for the sum of the series \(\sum_{n=0}^{\infty}(-5)^{n} .\) Show that this series is divergent. Do you see what happened? Warning hint: Always consider whether an answer is reasonable, whether it's a computer answer or your work by hand.

Short answer

The series \(\sum_{n=0}^{\infty}(-5)^{n}\) is divergent because \(\lim_{n \to \infty} (-5)^n\) does not approach zero.

Step-by-step solution

Define the Series

The given series is \(\textstyle \sum_{n=0}^{\infty}(-5)^{n}\).

Identify the General Term

The general term of the series is \(a_n = (-5)^n\).

Apply the Divergence Test

To determine if the series \(\textstyle \sum a_n\) is convergent or divergent, check the limit of the general term \(a_n\) as \(n\) approaches infinity. \(\textstyle \lim_{n \to \infty}a_n = \lim_{n \to \infty} (-5)^n \).

Evaluate the Limit

Since \(-5\) is a constant and not within the interval \(-1, 1\), \((-5)^n\) grows exponentially as \(n\) increases. Hence, \(\lim_{n \to \infty} (-5)^n = \infty\) or \(-\big.\infty\), depending on whether \(n\) is even or odd.

Conclusion from Divergence Test

Since \(\lim_{n \to \infty} a_n \) does not approach zero, the series \(\textstyle \sum_{n=0}^{\infty}(-5)^{n}\) is divergent.

Analyze the Given Result

The result given by the computer program, \(\frac{1}{6}\), is not reasonable because it violates the divergence test results. The series \(\textstyle \sum_{n=0}^{\infty}(-5)^{n}\) is divergent and should not sum to any finite value.

Key concepts

Divergence Test

One of the core strategies in determining whether an infinite series converges or diverges is the Divergence Test. This method hinges on a simple yet essential concept: If the limit of the general term of the series as it approaches infinity does not equal zero, the series cannot possibly converge.
For a series \(\textstyle \sum_{n=0}^{\infty} a_n\) to be convergent, it's necessary (but not sufficient) that \(\textstyle \lim_{n \to \infty}a_n = 0\). If this limit is not zero, the series must diverge.
Let's apply it to our example: the series \(\textstyle \sum_{n=0}^{\infty}(-5)^{n}\). We look at the general term \((a_n = (-5)^n)\). As n approaches infinity, the term \((-5)^n\) grows exponentially, and does not tend to zero. Hence, the series \(\textstyle \sum_{n=0}^{\infty}(-5)^{n}\) diverges.

Infinite Series

Understanding infinite series is key in calculus and beyond. An infinite series is the sum of the terms of an infinite sequence. It's written in the form \(\textstyle \sum_{n=0}^{\infty} a_n\). Each term in the series follows a specific pattern or rule.
Infinite series are captivating for their properties and their wide applications in mathematics, physics, and engineering. They can either converge to a specific value or diverge, implying they don’t sum up to a finite result.
Convergence and divergence of series are essential aspects to study. For instance, if a series converges, its partial sums \(\textstyle S_n = \sum_{k=0}^{n}a_k\) approach a finite value as n approaches infinity. On the flip side, if these partial sums do not approach a finite value, the series is said to diverge.

Exponential Growth

Exponential growth is a concept where quantities increase rapidly over time. It holds special significance in infinite series and is crucial for understanding why certain series diverge.
In mathematics, exponential growth refers to sequences where the factor of growth is consistently multiplied, such as the term \((-5)^n\). As n increases, this term grows exponentially.
When dealing with the series \(\textstyle \sum_{n=0}^{\infty}(-5)^{n}\), each additional term becomes exponentially larger since \((-5)^n\) vastly increases in magnitude as n increases. This exponential growth ensures that \((-5)^n\) doesn't settle towards zero, leading to the divergence of the series.

• Exponential growth is characterized by rapid escalation, often depicted as a swiftly rising curve.
• This rapid increase explains the infinite nature of many divergent series such as the one in the exercise.