Problem 31 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=2+(0.1)^{n} $$

Step-by-Step Solution

Answer

The sequence converges; its limit is 2.

Step 1: Analyze the General Term Behavior

Observe the general term of the sequence: \(a_n = 2 + (0.1)^n\). The sequence consists of two parts: a constant term 2 and a term \((0.1)^n\) that changes with \(n\). Here, \(0.1\) is a positive fraction less than 1.

Step 2: Determine the Convergence of \((0.1)^n\)

As \(n\) increases, \((0.1)^n\) becomes increasingly smaller because \(0.1 < 1\). In fact, the term \((0.1)^n\) approaches 0 as \(n\) approaches infinity. Mathematically, this is shown as:\[\lim_{{n \to \infty}} (0.1)^n = 0.\]

Step 3: Determine the Behavior of the Whole Sequence

Given that \((0.1)^n\) approaches 0, we apply this knowledge to the entire sequence:\[\lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} (2 + (0.1)^n) = 2 + 0 = 2.\]This indicates that as \(n\) approaches infinity, \(a_n\) approaches 2.

Step 4: Conclusion on Convergence

Because the limit \(\lim_{{n \to \infty}} a_n = 2\) exists and is finite, the sequence \(a_n = 2 + (0.1)^n\) is convergent. The limit of the sequence is 2.

Key Concepts

Limit of a Sequence

The limit of a sequence is a concept that helps to determine the behavior of a sequence as its index, usually denoted as \(n\), approaches infinity. In simpler terms, it's about finding what value the terms of the sequence get closer to as \(n\) becomes very large. For the given sequence \(a_n = 2 + (0.1)^n\), the main task is to see what happens to \((0.1)^n\) as \(n\) gets bigger.

• As \(n\) increases, \((0.1)^n\) shrinks because it's multiplying 0.1 by itself repeatedly, getting closer to zero.
• When something approaches zero, like \((0.1)^n\), our interest is in how it affects the entire sequence \(a_n\).
• The constant part 2 of the sequence remains unaffected by \(n\) and stays the same.

So, as \((0.1)^n\) gets closer to zero, the whole sequence \(a_n = 2 + (0.1)^n\) approaches 2, which is the limit of the sequence.

Convergence and Divergence

Convergence and divergence describe how the terms of a sequence behave as \(n\) increases. A sequence converges if it approaches a specific number, known as the limit, while a sequence diverges if it doesn't get close to a single value.
For the sequence \(a_n = 2 + (0.1)^n\):

• The component \((0.1)^n\) converges to 0 as \(n\) increases.
• Adding zero to 2 means that \(a_n\) converges to 2.
• Thus, the sequence \(a_n = 2 + (0.1)^n\) converges.

Understanding convergence helps us know that the sequence will settle down around a particular value, in this case, the number 2. This is why it's described as convergent, as opposed to diverging infinitely or oscillating without end.

Mathematical Analysis

Mathematical analysis involves using rigorous methods to study concepts like limits, convergence, and divergence. It's about making sure our understanding and calculations about sequences are correct and precise.
In our sequence \(a_n = 2 + (0.1)^n\):

• We carefully observe that \((0.1)^n\) shrinks towards zero as \(n\) gets larger.
• This observation allows us to determine the behavior of \(a_n\) by adding 2 to this diminishing term.
• Through analysis, we confidently conclude that the sequence converges with a limit of 2.

Think of mathematical analysis as a toolbox. It helps you to unpack complex behavior into understandable parts and verify results like whether a sequence converges or diverges. Using logical reasoning and mathematics, like limits and summation, makes our findings reliable.