Question

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$ a_{n}=\left(1-\frac{1}{n}\right)^{n^{2}} $$

Short answer

The series converges by the root test.

Step-by-step solution

Choose the Test

To determine which test is suitable, note that the sequence includes exponential terms. The ratio test might not be effective for exponential terms, so let's use the root test.

Apply the Root Test Formula

The root test involves computing the limit:\[ L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \]Substitute \(a_n = \left(1 - \frac{1}{n}\right)^{n^2}\) into this formula.

Simplify the Expression

Evaluate \( \sqrt[n]{\left(1 - \frac{1}{n}\right)^{n^2}} \):\[ |a_n|^{1/n} = \left(\left(1 - \frac{1}{n}\right)^{n^2}\right)^{1/n} = \left(1 - \frac{1}{n}\right)^{n} \]

Calculate the Limit

Find the limit as \(n\) approaches infinity:\[ L = \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{n} \]From known limits, this expression approaches \( e^{-1} = \frac{1}{e} \).

Interpret the Result

Since \(L = \frac{1}{e} < 1\), according to the root test, the series \(\sum a_{n}\) converges.

Key concepts

Understanding the Root Test

The root test is an essential tool for determining the convergence of an infinite series.It is especially helpful when dealing with infinite series involving exponential expressions or complicated powers. The root test requires calculating the n-th root of the absolute value of the sequence's terms. To apply this test, consider the limit:\[L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\]

• If \(L < 1\), then the series converges.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive.

One of the significant advantages of the root test is its ability to handle sequences with exponential factors efficiently. When applied correctly, as in the solved step where \(a_n = \left(1 - \frac{1}{n}\right)^{n^2}\), it shows that the series converges since the limit evaluates to \(\frac{1}{e}\). This value is less than 1, confirming convergence by the test.

Utilizing the Ratio Test

The ratio test is another method for checking the convergence of infinite series. It proves highly effective for sequences where terms are ratios of polynomials or factorials. To employ this test, compute the limit of the absolute ratio of successive terms:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]

• If \(L < 1\), the series converges absolutely.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive, and another method must be used.

The ratio test provides a straightforward and often quick determination of convergence. It can be particularly useful when dealing with series involving factorials or simple exponential terms. However, for complex exponential functions, like in our given series, the root test is generally more suitable. Choosing between the ratio and root tests usually depends on the form of the series.

Exploring Infinite Series

An infinite series is the sum of the terms of an infinite sequence.Studying these series is crucial because they frequently emerge in mathematical analysis, physics, and engineering.The concept of convergence is central to infinite series, determining whether the sum approaches a finite number.In a convergent series, as more terms are added, the sum approaches a specific finite value.Meanwhile, a divergent series doesn't approach any finite limit.Understanding how an infinite series behaves requires testing for convergence using tools such as the root and ratio tests.Convergence tests help ascertain not only whether a series converges or diverges but also the nature of its convergence.Employing convergence tests, like in the solution for the series with terms \(a_n = \left(1 - \frac{1}{n}\right)^{n^2}\), is essential for deeper insights into the behavior of mathematical expressions and functions.