Question

Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ a_{n}=n^{2} / 2^{n} $$

Short answer

The series converges as the limit of the ratio is less than 1.

Step-by-step solution

Write out the formula for the Ratio Test

The ratio test is used to determine the convergence of a series \( \sum_{n=1}^{\infty} a_n \). For a series \( \sum a_n \), the ratio test considers the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If this limit is less than 1, the series converges; if it's greater than 1, the series diverges; if it equals 1, the test is inconclusive.

Express \(a_{n+1}\) and write the fraction \(\frac{a_{n+1}}{a_n}\)

Given \( a_n = \frac{n^2}{2^n} \), we first need to express \( a_{n+1} \). Thus, \( a_{n+1} = \frac{(n+1)^2}{2^{n+1}} \). Now, find \( \frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)^2}{2^{n+1}}}{\frac{n^2}{2^n}} = \frac{(n+1)^2}{n^2} \cdot \frac{1}{2} \).

Simplify the expression \(\frac{a_{n+1}}{a_n}\)

Simplify \( \frac{(n+1)^2}{n^2} \cdot \frac{1}{2} \). This gives \( \frac{(n+1)^2}{2n^2} = \frac{n^2 + 2n + 1}{2n^2} \). The simplified form is\( \frac{1}{2} + \frac{1}{n} + \frac{1}{2n^2} \).

Calculate the limit as \(n\) approaches infinity

We now need to calculate \( \lim_{n \to \infty} \left( \frac{1}{2} + \frac{1}{n} + \frac{1}{2n^2} \right) \). As \( n \) approaches infinity, \( \frac{1}{n} \) and \( \frac{1}{2n^2} \) both approach 0. Thus, the limit becomes \( \frac{1}{2} \).

Interpret the result of the Ratio Test Limit

Since the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{2} \) is less than 1, the ratio test indicates that the series \( \sum_{n=1}^{\infty} \frac{n^2}{2^n} \) converges.

Key concepts

Series Convergence

Convergence is a fundamental idea in series and sequences where terms get closer to a particular value as we continue further along the series. For a series like \( \sum_{n=1}^{\infty} a_n \), determining convergence means finding out whether the total sum approaches a fixed number as the number of terms increases.
In practice, if a series converges, adding more terms makes the sum approach closer to the limit. On the contrary, divergence implies the sum doesn't approach a fixed limit and often continues to increase or decrease without bound.

• If a series converges, we can consider it as a reliable estimate for its limit after a certain point.
• A divergent series indicates that the series does not stabilize at a particular value.

In this exercise, understanding convergence helps us determine whether the given series stabilizes to a specific limit as more terms are added.

Limit Calculation

Finding limits is crucial when analyzing series or functions. A limit examines the behavior of a function as an input approaches a certain value, usually infinity in series context. Here, we use limits to analyze terms' behavior in a series as the index \( n \) becomes infinitely large.

Why Calculating the Limit Matters

Calculating limits helps determine the eventual behavior of series terms when applying tests like the Ratio Test. It tells us if terms shrink down towards zero sufficiently fast, which is often necessary for convergence.

• As \( n \to \infty \), the term contributions from parts like \( \frac{1}{n} \) reduce to zero.
• The limit \( \lim_{n \to \infty} \left( \frac{1}{2} + \frac{1}{n} + \frac{1}{2n^2} \right) \) simplifies the decision about convergence.

Using limits, we efficiently decide if the sum accumulates to a stable number or not after enough terms are included.

Infinite Series

Understanding infinite series involves dealing with sums of infinitely many terms. Typically, each term comes from a sequence defined by some mathematical rule. These concepts allow us to find out whether summing an infinite number of elements can yield a finite result.
In mathematical analysis, working with infinite series often requires applying specific tests to assess convergence.

• An infinite series may converge if adding up infinitely many terms approaches a finite value.
• If summing the series' terms doesn't approach any particular limit, the series diverges.

In this problem, the series \( \sum_{n=1}^{\infty} \frac{n^2}{2^n} \) represents an example of an infinite series, which needs convergence testing to understand its behavior.

Mathematical Tests

Mathematical tests are tools used to analyze the behavior and convergence of series. Among these, the Ratio Test is a popular method used for series whose terms include factorial or exponential components.

How the Ratio Test Works

The Ratio Test computes the limit of the ratio of successive terms. Specifically, for a series \( \sum_{n=1}^{\infty} a_n \), we evaluate \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

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

Applying the Ratio Test to \( \sum_{n=1}^{\infty} \frac{n^2}{2^n} \), we find that the limit calculation leads to \( \frac{1}{2} \), suggesting convergence because \( \frac{1}{2} < 1 \). The series converges, confirming the usefulness of the Ratio Test in determining series behavior efficiently.