Problem 37

Question

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{3^{n-1}+1}\end{equation}

Step-by-Step Solution

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Answer

The series $ \sum_{n=1}^{\infty} \frac{1}{3^{n-1}+1} $ converges.

1Step 1: Identify a Suitable Convergence Test

To determine the convergence or divergence of the series $ \sum_{n=1}^{\infty} \frac{1}{3^{n-1}+1} $, we will use the Limit Comparison Test. This test is suitable here because $ \frac{1}{3^{n-1}+1} $ resembles a geometric series which is a well-known comparison series.

2Step 2: Choose a Comparison Series

We select the geometric series $ \sum_{n=1}^{\infty} \frac{1}{3^n} $ to compare with our series. This geometric series has a common ratio $ r = \frac{1}{3} $, where $ |r| < 1 $, so it converges.

3Step 3: Apply the Limit Comparison Test

To apply the Limit Comparison Test, compute the limit $ L = \lim_{n \to \infty} \frac{a_n}{b_n} $ where $ a_n = \frac{1}{3^{n-1}+1} $ and $ b_n = \frac{1}{3^n} $. Simplify and compute:\[L = \lim_{n \to \infty} \frac{\frac{1}{3^{n-1}+1}}{\frac{1}{3^n}} = \lim_{n \to \infty} \frac{3^n}{3^{n-1} + 1} = \lim_{n \to \infty} \frac{3}{1 + \frac{1}{3^{n-1}}} = 3\]The limit $ L = 3 $ which is positive and finite.

4Step 4: Conclude with the Limit Comparison Test

Since the limit $ L = 3 $ is a positive finite number, and the comparison series $ \sum_{n=1}^{\infty} \frac{1}{3^n} $ converges, the Limit Comparison Test tells us that the series $ \sum_{n=1}^{\infty} \frac{1}{3^{n-1}+1} $ also converges.

Key Concepts

limit comparison testgeometric seriesseries convergenceinfinite series

limit comparison test

The Limit Comparison Test is a powerful tool for determining the convergence or divergence of an infinite series. It's often used when a series resembles another, more familiar series whose convergence properties are known. The core idea is to compare the original series to a simpler one by taking the limit of their ratio.
Let’s break it down:

Choose two series, for instance, $ \sum a_n $ and $ \sum b_n $, where $ b_n $ is a series with known convergence.
Compute the limit $ L = \lim_{n \to \infty} \frac{a_n}{b_n} $.

If the limit $ L $ is a positive finite number, then both series will either converge or diverge together. It is crucial to select a comparison series with similar behavior to ensure the test reflects true convergent properties.

geometric series

A geometric series is a series of the form $ \sum_{n=0}^{\infty} ar^n $, where $ a $ is the first term and $ r $ is the common ratio. The series solely depends on the value of $ |r| $ for its convergence:

If $ |r| < 1 $, the series converges to the sum $ \frac{a}{1-r} $.
If $ |r| \geq 1 $, the series diverges.

A notable advantage of geometric series is their simplicity and the straightforward conditions which dictate their convergence. This makes them a popular choice for comparison in convergence tests, such as the Limit Comparison Test, especially when the series in question takes a simple exponential form.

series convergence

Series convergence is crucial in the study of infinite series. It refers to whether the sum of an infinite series settles down to a finite number or not. There are several tests available to determine convergence, with each test suitable for different scenarios:

Limit Comparison Test: Ideal when comparing to a geometric or p-series.
Ratio Test: Useful for series involving factorials or exponentials.
Integral Test: Particularly handy when the series terms are positive and resemble a function that can be integrated.

Understanding when and how to apply these tests can help you identify the convergence of a series effectively. Each test has specific criteria and conditions under which they yield correct conclusions about convergence.

infinite series

An infinite series is the sum of an infinite sequence of terms. Formally, it can be expressed as $ \sum_{n=1}^{\infty} a_n $, where $ a_n $ are the terms of the sequence. The key interest in examining such series is to determine whether they converge to a finite value or diverge to infinity or somewhere undefined.
Considerations include:

Convergence Tests: Techniques like the Limit Comparison, Ratio, and Integral tests help determine the behavior of these sums.
Types of Series: Includes geometric series, p-series, harmonic series, etc.
Application: Convergent series find applications in calculus, physics, and finance for approximating complex phenomena.

Understanding infinite series requires practice, patience, and learning to choose appropriate tests and series for comparison.

Problem 37

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