Problem 1 - Calculus

Question

What is \(a_{4}\) and what is the limit \(L\) ? After which \(N\) is \(\left|a_{n}-L\right|<\frac{1}{10} ?\) (Calculator allowed) (a) \(-1,+\frac{1}{2},-\frac{1}{3}, \ldots\) (b) \(\frac{1}{2}, \frac{1}{2}+\frac{1}{4}, \frac{1}{2}+\frac{1}{4}+\frac{1}{6}, \ldots\) (c) \(\frac{1}{2}, \frac{2}{4}, \frac{3}{8}, \ldots a_{n}=n / 2^{n}\) (d) \(1.1,1.11,1.111, \ldots\) (e) \(a_{n}=\sqrt[n]{n}\) (f) \(a_{n}=\sqrt{n^{2}+n}-n\) (g) \(1+1,\left(1+\frac{1}{2}\right)^{2},\left(1+\frac{1}{3}\right)^{3}, \ldots\)

Step-by-Step Solution

Answer

(a) No limit. (b) No limit. (c) \(L=0\), \(N=7\). (d) No limit. (e) \(L=1\), \(N=10\). (f) \(L=\frac{1}{2}\), \(N=11\). (g) \(L=e\), numerical exploration for \(N\).

Step 1: Analyze Sequence (a)

The sequence is given as \(-1, \frac{1}{2}, -\frac{1}{3}, \ldots\). The pattern is alternating signs with terms of the form \(a_n = \frac{(-1)^n}{n}\). To find the fourth term \(a_4\), substitute \(n = 4\): \(a_4 = \frac{(-1)^4}{4} = \frac{1}{4}\). This is a divergent series, so no specific limit \(L\) exists.

Step 2: Analyze Sequence (b)

The sequence \(\frac{1}{2}, \frac{1}{2}+\frac{1}{4}, \frac{1}{2}+\frac{1}{4}+\frac{1}{6}, \ldots\) is partial sums of the harmonic series starting from \(\frac{1}{2}\). To find \(a_4\), confirm the pattern: \(a_4 = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} = 1.2083\) (approx). This is a divergent series without a specific limit \(L\).

Step 3: Analyze Sequence (c)

The sequence \(\frac{1}{2}, \frac{2}{4}, \frac{3}{8}, \ldots\) is defined by \(a_n = \frac{n}{2^n}\). For \(a_4\), calculate: \(a_4 = \frac{4}{16} = \frac{1}{4}\). To find the limit, we note that \(\lim_{n \to \infty} \frac{n}{2^n} = 0\). The sequence approaches 0, so \(L = 0\). We need \(n\) such that \(\left| a_n - 0 \right| < \frac{1}{10}\), solve \(\frac{n}{2^n} < \frac{1}{10}\). Through iteration, find \(N = 7\) as smallest value satisfying the condition.

Step 4: Analyze Sequence (d)

The sequence \(1.1, 1.11, 1.111, \ldots\) is a construction where each term adds an extra 1 in the next decimal place. \(a_4 = 1.1111\). There is no limit \(L\) as the sequence continues to increase without bound.

Step 5: Analyze Sequence (e)

Given \(a_n = \sqrt[n]{n}\), find \(a_4\): \(a_4 = \sqrt[4]{4} = 4^{1/4} = \sqrt{2} \approx 1.41\). To find the limit, note that as \(n\) approaches infinity, \(\sqrt[n]{n} \to 1\). Therefore, \(L = 1\). Find \(N\) such that \(\left| \sqrt[n]{n} - 1 \right| < \frac{1}{10}\). Solve through iteration to find \(N = 10\).

Step 6: Analyze Sequence (f)

For \(a_n = \sqrt{n^2 + n} - n\), calculate \(a_4\). It simplifies to \(\sqrt{20} - 4 \approx 0.47\). As \(n\) increases, \(a_n\) approaches a limit \(L = \frac{1}{2}\) since \(\sqrt{n^2 + n} \sim n + \frac{1}{2}\) as \(n\) becomes very large. Determine \(N\) such that \(\left|a_n - \frac{1}{2}\right| < \frac{1}{10}\). Solving similar to before, we find \(N = 11\).

Step 7: Analyze Sequence (g)

The sequence \(1+1, \left(1+\frac{1}{2}\right)^2, \left(1+\frac{1}{3}\right)^3, \ldots\) follows \(a_n = \left(1+\frac{1}{n}\right)^n\), approaching \(e\) (around 2.718) as \(n\) goes to infinity. For \(a_4\), calculate \(\left(1+\frac{1}{4}\right)^4 = \approx 2.441\). \(L = e\). Find \(N\) for \(\left|a_n - e\right| < \frac{1}{10}\) which requires numerical exploration; often beyond standard calculators, use approximations.

Key Concepts

Divergent Series

A series is considered divergent when it does not have a finite limit as more terms are added to it. This means even as you continue adding numbers in the sequence, you don't converge to a specific value. For instance, in exercise part (a), the sequence

• consists of alternating positive and negative terms that decrease in magnitude.
• The sequence formula is given by \(a_n = \frac{(-1)^n}{n}\).
• Here, the series does not settle at a single value, making it divergent.

Divergent series are common in mathematical analysis and determining their behavior involves checking summation patterns, and recognizing conditions that cause divergence.

Harmonic Series

The harmonic series is a classic example in mathematical series which illustrates divergence, even though its terms grow very slowly. In exercise part (b), we look at the partial sums of a harmonic series starting from \(\frac{1}{2}\). This gives us

• sequences such as \(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots\).
• Even though it initially looks like it might converge, the sums increase without bound.

This illustrates that smaller and smaller positive numbers still contribute enough to prevent convergence. Hence, the harmonic series is divergent. Engineers and scientists sometimes encounter harmonic-like series in calculations where the sum needs consideration, regardless of its divergence.

Limits of Sequences

The limit of a sequence is the value the terms of the sequence approach as the term number goes towards infinity. In part (c) of the exercise,

• the sequence \(a_n = \frac{n}{2^n}\) has a pattern where each term shrinks closer to zero as \(n\) increases.
• This behavior demonstrates convergence towards a specific limit.
• In this case, the limit \(L\) is 0 because no matter how small \(\epsilon\) you choose, there exists an \(N\) such that for all \(n > N\), the terms stay within \(\epsilon\) of 0.

Understanding limits is crucial in calculus and analysis, guiding how we predict the long-term behavior of sequences.

Asymptotic Behavior

Asymptotic behavior describes how sequences or functions behave as they get increasingly larger or extend towards infinity. For example, in sequence part (f),

• the asymptotic expression \(a_n = \sqrt{n^2+n} - n\) simplifies to approach \(\frac{1}{2}\) as \(n\) gets very large.
• This simplification is important for understanding long-term growth and remaining differences as the sequence grows.
• The analysis often involves looking beyond the immediate pattern to see how changes in the sequence happen over broader scales.

Studying asymptotic behavior is useful in both mathematical theory and real-world applications for predicting outcomes and systems behaviors over long periods or extensive scales.

Convergence Criteria

Convergence criteria are rules or tests used to determine whether a series or sequence will converge to a limit. In exercise part (g), we explored convergence through the series \(a_n = \left(1 + \frac{1}{n}\right)^n\), which approaches \(e\) as \(n\) increases. To understand convergence, consider:

• Absolute Convergence: If the series of absolute values converges, then the original series converges.
• Using tests like the Limit Test, Ratio Test, or Integral Test can help establish convergence.
• Finding \(N\) that satisfies \(\left| a_n - L \right| < \epsilon\) for a small \(\epsilon\) gives practical ways to quantify when convergence is achieved.

These criteria are essential tools for mathematicians and engineers who need to guarantee precise results from mathematical series.