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How to determine if $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$ converges or diverges using the Direct Comparison Test?
1. Recognize $\frac{1}{n^2+1} < \frac{1}{n^2}$. 2. Know that $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges (p-series with p=2 > 1). 3. Conclude that $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$ converges by the Direct Comparison Test.
How to determine if $\sum_{n=1}^{\infty} \frac{n}{n^3-2}$ converges or diverges using the Limit Comparison Test?
1. Choose $b_n = \frac{n}{n^3} = \frac{1}{n^2}$. 2. Evaluate $\lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty} \frac{\frac{n}{n^3-2}}{\frac{1}{n^2}} = 1$. 3. Since the limit is finite and positive, and $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, conclude that $\sum_{n=1}^{\infty} \frac{n}{n^3-2}$ converges.
How to choose a comparison series $b_n$ for $\sum_{n=1}^{\infty} \frac{2n+1}{n^2+n+1}$?
1. Focus on the dominant terms: $2n$ in the numerator and $n^2$ in the denominator. 2. Form $b_n = \frac{2n}{n^2} = \frac{2}{n} = \frac{1}{n}$.
How to determine if $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} - 1}$ diverges?
1. Compare to $\frac{1}{\sqrt{n}}$. 2. Note that $\frac{1}{\sqrt{n}-1} > \frac{1}{\sqrt{n}}$. 3. Recognize that $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ diverges (p-series with p=1/2 < 1). 4. Conclude that $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} - 1}$ diverges by the Direct Comparison Test.
How to handle a series with a sine function in the numerator, such as $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^2}$?
1. Use the fact that $-1 \le \sin(n) \le 1$. 2. Compare to $\sum_{n=1}^{\infty} \frac{1}{n^2}$. 3. Since $\left|\frac{\sin(n)}{n^2}\right| \le \frac{1}{n^2}$ and $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, conclude that $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^2}$ converges absolutely by the Direct Comparison Test.
Given $\sum_{n=1}^{\infty} \frac{1}{2^n + n}$, how do you select a suitable $b_n$?
1. Notice that $2^n$ grows faster than $n$. 2. Choose $b_n = \frac{1}{2^n}$. 3. Use the Direct Comparison Test since $\frac{1}{2^n + n} < \frac{1}{2^n}$.
How to determine if $\sum_{n=2}^{\infty} \frac{1}{n\ln(n)}$ diverges?
1. Recognize that this is not directly comparable to a p-series or geometric series. 2. Consider the Integral Test (not a comparison test, but relevant). 3. Since $\int_2^{\infty} \frac{1}{x\ln(x)} dx$ diverges, conclude that $\sum_{n=2}^{\infty} \frac{1}{n\ln(n)}$ diverges.
How to determine if $\sum_{n=1}^{\infty} \frac{3^n}{4^n - 2^n}$ converges?
1. Compare to $b_n = \frac{3^n}{4^n} = (\frac{3}{4})^n$. 2. Use the Limit Comparison Test. 3. Since $\sum_{n=1}^{\infty} (\frac{3}{4})^n$ converges (geometric series with |r| < 1), conclude that $\sum_{n=1}^{\infty} \frac{3^n}{4^n - 2^n}$ converges.
How do you know when to use the Direct Comparison Test vs. the Limit Comparison Test?
Direct Comparison Test: when you can easily show $a_n < b_n$ or $a_n > b_n$. Limit Comparison Test: when it's difficult to find a direct inequality, but the limit of the ratio is easy to compute.
What is the first step in determining convergence/divergence using comparison tests?
Identify a suitable comparison series ($b_n$) with known convergence/divergence behavior.
Explain the purpose of comparison tests.
To determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known.
Why must $a_n$ and $b_n$ be non-negative when using comparison tests?
To ensure that the comparison is valid. If terms are negative, the inequalities used in the tests may not hold.
Explain the intuition behind the Direct Comparison Test.
If a larger series converges, a smaller series must also converge. If a smaller series diverges, a larger series must also diverge.
Explain the intuition behind the Limit Comparison Test.
If two series have similar end behavior, they will either both converge or both diverge.
What does the value of p tell you about the convergence of a p-series?
If $p > 1$, the p-series converges. If $p \le 1$, the p-series diverges.
What does the value of r tell you about the convergence of a geometric series?
If $|r| < 1$, the geometric series converges. If $|r| \ge 1$, the geometric series diverges.
How do you choose an appropriate series to compare to?
Choose a series that has similar terms and known convergence/divergence, often a p-series or geometric series.
When is the Direct Comparison Test most useful?
When it is easy to establish a direct inequality between the terms of the two series.
When is the Limit Comparison Test most useful?
When it is difficult to establish a direct inequality, but the limit of the ratio of the terms is easy to compute.
If $\lim_{n\to\infty} \frac{a_n}{b_n} = 0$, what does this imply?
That $b_n$ grows much faster than $a_n$.
Define Direct Comparison Test.
Compares a series to another known series to determine convergence/divergence. If $0 le a_n le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $\sum a_n$ diverges, then $\sum b_n$ diverges.
Define Limit Comparison Test.
Compares the limit of the ratio of two series terms. If $\lim_{n\to\infty} \frac{a_n}{b_n} = c$, where $0 < c < \infty$, then both series either converge or diverge.
Define Convergence.
A series converges if the sequence of its partial sums approaches a finite limit.
Define Divergence.
A series diverges if the sequence of its partial sums does not approach a finite limit.
Define p-series.
A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p$ is a real number.
Define Geometric Series.
A series of the form $\sum_{n=0}^{\infty} ar^n$, where $a$ is a constant and $r$ is the common ratio.
What is a series?
The sum of the terms of a sequence.
Define $a_n$ and $b_n$ in the context of comparison tests.
$a_n$ and $b_n$ are the terms of the two series being compared. They must be non-negative for comparison tests to be valid.
What does it mean for a limit to be 'finite'?
A finite limit is a real number (not infinity).
Define 'end behavior' in the context of series.
How the terms of a series behave as $n$ approaches infinity.