Question

Determine the convergence of the given series using the Root Test. If the Root Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right)^{n}$$

Short answer

The series converges absolutely by the Root Test.

Step-by-step solution

Understand the Root Test

The Root Test, also known as the nth root test, investigates the convergence of a series \( \sum a_n \) by examining the limit \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). If this limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive.

Define the nth Term

For the given series, the nth term \( a_n \) is \( \left( \frac{1}{n} - \frac{1}{n^2} \right)^n \). This will be used to evaluate the limit in the Root Test.

Apply the Root Test

Calculate \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{\left( \frac{1}{n} - \frac{1}{n^2} \right)^n} \). This simplifies to \( \lim_{n \to \infty} \left( \frac{1}{n} - \frac{1}{n^2} \right) \).

Simplify the Limit

As \( n \to \infty \), \( \frac{1}{n^2} \) becomes negligible compared to \( \frac{1}{n} \), so \( \frac{1}{n} - \frac{1}{n^2} \approx \frac{1}{n} \). Therefore, the limit becomes \( \lim_{n \to \infty} \frac{1}{n} = 0 \).

Determine Convergence

Since \( 0 < 1 \), by the Root Test, the series \( \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n^2} \right)^n \) converges absolutely.

Key concepts

nth root test

The nth Root Test, or simply the Root Test, is an essential tool in determining the convergence of series. Convergence tells us whether a series adds up to a finite number as you approach infinity. The nth Root Test helps find this out by focusing on the limit of the nth root of the absolute value of the terms of the series. If you have a series represented by \( \sum a_n \), you compute the limit \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). This means you're looking at the n-th root of the absolute value of each term in the series.

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

A limit less than 1 suggests that the series terms are getting smaller fast enough, leading to convergence. If above 1, they aren't diminishing quick enough, causing divergence. The test gives no clear answer if the result is exactly 1, so another test might be necessary.

series convergence

Series convergence is a central idea in calculus and analysis. It hints at whether the series summates to a finite value or diverges away as the series extends to infinity.When we look at a series such as \( \sum_{n=1}^{\infty} a_n \), we want to identify if the infinite process of summing up all terms leads to a definitive number. Tools like the nth Root Test help check for this convergence.Here, for example, after applying the Root Test, if we find the limit \( \lim_{n \to \infty} \sqrt[n]{|a_n|} < 1 \), the series converges absolutely. This means that the magnitudes of the series terms shrink to zero quickly enough to ensure a total sum. However, not all series converge absolutely; sometimes, absolute convergence isn't achieved, but that doesn't automatically imply divergence.This is where understanding series properties and using the correct tests is key in understanding convergence or divergence behavior.

absolute convergence

Absolute convergence refers to whether a series converges when you take the absolute values of its terms. This condition is stricter than general convergence because it guarantees that terms are small and cancel out any potential divergent behavior.In formal terms, a series \( \sum a_n \) is absolutely convergent if the series constructed by the absolute values, \( \sum |a_n| \), also converges. This stricter form of convergence is vital when dealing with series that include negative terms, ensuring that fluctuations in sign don't affect overall convergence.The Root Test evaluates absolute convergence by checking if the nth root of the terms diminishes fast enough. If the series converges absolutely, it certainly converges normally. However, a series that doesn't converge absolutely might still converge due to alternating positive and negative terms properly canceling each other out.

limit evaluation

Limit evaluation is intrinsic to analyzing series using the Root Test. The process involves examining the behavior of terms as they approach infinity. Evaluating the limit helps us understand how smaller terms need to become for a series to converge.In our series \( \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n^2} \right)^n \), we specifically assess the limit \( \lim_{n \to \infty} \left( \frac{1}{n} - \frac{1}{n^2} \right) \). As \( n \to \infty \), the fractional term \( \frac{1}{n^2} \) becomes negligible. Hence, the sequence simplifies remarkably to \( \frac{1}{n} \), showing that the terms of the series converge to zero as needed for convergence.Proper evaluation of limits provides concrete insights into whether a series will add up to a finite quantity. It allows us to apply convergence tests effectively, predicting series behavior far down the end of their progressions.