Comparison Tests for Convergence - AP Calculus BC
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Use Comparison: is $\frac{1}{n^2+2n}$ convergent?
Use Comparison: is $\frac{1}{n^2+2n}$ convergent?
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Converges; compare to $\frac{1}{n^2}$ (convergent $p$-series). For large $n$, $\frac{1}{n^2+2n} \sim \frac{1}{n^2}$.
Converges; compare to $\frac{1}{n^2}$ (convergent $p$-series). For large $n$, $\frac{1}{n^2+2n} \sim \frac{1}{n^2}$.
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Does $\frac{1}{n}$ converge? Use the Comparison Test.
Does $\frac{1}{n}$ converge? Use the Comparison Test.
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Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). $\frac{1}{n}$ is a divergent $p$-series with $p = 1$.
Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). $\frac{1}{n}$ is a divergent $p$-series with $p = 1$.
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Identify a series to compare with $\frac{1}{n^4+3n^2}$.
Identify a series to compare with $\frac{1}{n^4+3n^2}$.
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Compare with $\frac{1}{n^4}$, a convergent $p$-series. For large $n$, $\frac{1}{n^4+3n^2} \sim \frac{1}{n^4}$.
Compare with $\frac{1}{n^4}$, a convergent $p$-series. For large $n$, $\frac{1}{n^4+3n^2} \sim \frac{1}{n^4}$.
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State the Limit Comparison Test.
State the Limit Comparison Test.
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If $\frac{a_n}{b_n} \to c > 0$, both series converge or diverge. Both series share the same convergence behavior when the limit exists and is positive.
If $\frac{a_n}{b_n} \to c > 0$, both series converge or diverge. Both series share the same convergence behavior when the limit exists and is positive.
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State the condition for the Limit Comparison Test to be valid.
State the condition for the Limit Comparison Test to be valid.
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If $\frac{a_n}{b_n} \to c > 0$, it is valid. This ensures both series have the same convergence behavior.
If $\frac{a_n}{b_n} \to c > 0$, it is valid. This ensures both series have the same convergence behavior.
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In the Comparison Test, what do you compare $a_n$ to?
In the Comparison Test, what do you compare $a_n$ to?
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Compare $a_n$ to $b_n$ where $b_n$ is a known series. The known series $b_n$ serves as the reference for comparison.
Compare $a_n$ to $b_n$ where $b_n$ is a known series. The known series $b_n$ serves as the reference for comparison.
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For Limit Comparison, what happens if $c = 0$ or $c = \text{infinity}$?
For Limit Comparison, what happens if $c = 0$ or $c = \text{infinity}$?
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The test is inconclusive if $c = 0$ or $c = \text{infinity}$. These limit values don't provide definitive comparison information.
The test is inconclusive if $c = 0$ or $c = \text{infinity}$. These limit values don't provide definitive comparison information.
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What is required for the Comparison Test to be applicable?
What is required for the Comparison Test to be applicable?
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Series terms $a_n$ and $b_n$ must be positive for all $n$. Negative terms invalidate the comparison inequalities used in the test.
Series terms $a_n$ and $b_n$ must be positive for all $n$. Negative terms invalidate the comparison inequalities used in the test.
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Use Comparison: is $\frac{n^2}{n^3+1}$ convergent?
Use Comparison: is $\frac{n^2}{n^3+1}$ convergent?
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Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). For large $n$, $\frac{n^2}{n^3+1} \sim \frac{1}{n}$.
Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). For large $n$, $\frac{n^2}{n^3+1} \sim \frac{1}{n}$.
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When using the Comparison Test, what must be true of $b_n$?
When using the Comparison Test, what must be true of $b_n$?
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$b_n$ must be a series with known convergence behavior. Without known behavior, no conclusion can be drawn.
$b_n$ must be a series with known convergence behavior. Without known behavior, no conclusion can be drawn.
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Determine convergence of $\frac{1}{n^{2.5}}$ using Comparison.
Determine convergence of $\frac{1}{n^{2.5}}$ using Comparison.
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Converges; compare to $\frac{1}{n^{2.5}}$ (convergent $p$-series). $p = 2.5 > 1$ makes this a convergent $p$-series.
Converges; compare to $\frac{1}{n^{2.5}}$ (convergent $p$-series). $p = 2.5 > 1$ makes this a convergent $p$-series.
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Use Comparison: is $\frac{1}{\text{ln}(n) \times n}$ convergent?
Use Comparison: is $\frac{1}{\text{ln}(n) \times n}$ convergent?
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Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). Since $\ln(n)$ grows slower than any power, this behaves like $\frac{1}{n}$.
Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). Since $\ln(n)$ grows slower than any power, this behaves like $\frac{1}{n}$.
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For Limit Comparison, what if $a_n$ and $b_n$ are non-positive?
For Limit Comparison, what if $a_n$ and $b_n$ are non-positive?
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Test is not applicable; terms must be positive. The test relies on positive term inequalities.
Test is not applicable; terms must be positive. The test relies on positive term inequalities.
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Use Limit Comparison: compare $\frac{n}{n^2+1}$ with $\frac{1}{n}$.
Use Limit Comparison: compare $\frac{n}{n^2+1}$ with $\frac{1}{n}$.
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Diverges; limit is finite and positive. $\lim_{n \to \infty} \frac{n}{n^2+1} \cdot \frac{n}{1} = 1 > 0$.
Diverges; limit is finite and positive. $\lim_{n \to \infty} \frac{n}{n^2+1} \cdot \frac{n}{1} = 1 > 0$.
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Use Comparison: does $\frac{n^2 + 3}{n^3 + 1}$ converge?
Use Comparison: does $\frac{n^2 + 3}{n^3 + 1}$ converge?
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Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). For large $n$, $\frac{n^2+3}{n^3+1} \sim \frac{1}{n}$.
Diverges; compare to $\frac{1}{n}$ (divergent $p$-series). For large $n$, $\frac{n^2+3}{n^3+1} \sim \frac{1}{n}$.
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Use Limit Comparison: compare $\frac{1}{n^2+1}$ with $\frac{1}{n^2}$.
Use Limit Comparison: compare $\frac{1}{n^2+1}$ with $\frac{1}{n^2}$.
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Converges; limit is finite and positive. $\lim_{n \to \infty} \frac{1}{n^2+1} \cdot \frac{n^2}{1} = 1 > 0$.
Converges; limit is finite and positive. $\lim_{n \to \infty} \frac{1}{n^2+1} \cdot \frac{n^2}{1} = 1 > 0$.
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Use Limit Comparison: compare $\frac{n^2+1}{n^4}$ with $\frac{1}{n^2}$.
Use Limit Comparison: compare $\frac{n^2+1}{n^4}$ with $\frac{1}{n^2}$.
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Converges; limit is finite and positive. $\lim_{n \to \infty} \frac{n^2+1}{n^4} \cdot \frac{n^2}{1} = 1 > 0$.
Converges; limit is finite and positive. $\lim_{n \to \infty} \frac{n^2+1}{n^4} \cdot \frac{n^2}{1} = 1 > 0$.
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Identify a series to compare with $\frac{3}{n^3 + \text{ln}(n)}$.
Identify a series to compare with $\frac{3}{n^3 + \text{ln}(n)}$.
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Compare with $\frac{1}{n^3}$, a convergent $p$-series. For large $n$, $\frac{3}{n^3 + \ln(n)} \sim \frac{3}{n^3}$.
Compare with $\frac{1}{n^3}$, a convergent $p$-series. For large $n$, $\frac{3}{n^3 + \ln(n)} \sim \frac{3}{n^3}$.
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Does the Limit Comparison Test require non-negative terms?
Does the Limit Comparison Test require non-negative terms?
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Yes, terms $a_n$ and $b_n$ must be positive. Positivity is essential for meaningful ratio comparison.
Yes, terms $a_n$ and $b_n$ must be positive. Positivity is essential for meaningful ratio comparison.
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What $p$-series can you use to compare when $p > 1$?
What $p$-series can you use to compare when $p > 1$?
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Use $\frac{1}{n^p}$, which converges for $p > 1$. $p$-series with $p > 1$ are standard convergent comparison series.
Use $\frac{1}{n^p}$, which converges for $p > 1$. $p$-series with $p > 1$ are standard convergent comparison series.
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Can the Limit Comparison Test be used if $c < 0$?
Can the Limit Comparison Test be used if $c < 0$?
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No, $c$ must be positive for the Limit Comparison Test. Negative limits don't maintain the required proportional relationship.
No, $c$ must be positive for the Limit Comparison Test. Negative limits don't maintain the required proportional relationship.
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What is the relationship between $a_n$ and $b_n$ in Limit Comparison?
What is the relationship between $a_n$ and $b_n$ in Limit Comparison?
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If $\frac{a_n}{b_n} \to c > 0$, both series behave similarly. When $c > 0$, the series have proportional terms and same convergence.
If $\frac{a_n}{b_n} \to c > 0$, both series behave similarly. When $c > 0$, the series have proportional terms and same convergence.
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Can the Comparison Test be used for alternating series?
Can the Comparison Test be used for alternating series?
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No, the test requires positive terms. Alternating series have both positive and negative terms.
No, the test requires positive terms. Alternating series have both positive and negative terms.
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Is $\frac{1}{n^{1.5}}$ convergent? Use Comparison.
Is $\frac{1}{n^{1.5}}$ convergent? Use Comparison.
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Converges; compare to $\frac{1}{n^{1.5}}$ (convergent $p$-series). $p = 1.5 > 1$ makes this a convergent $p$-series.
Converges; compare to $\frac{1}{n^{1.5}}$ (convergent $p$-series). $p = 1.5 > 1$ makes this a convergent $p$-series.
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Identify if $\frac{1}{n^3 + n}$ is convergent using Comparison.
Identify if $\frac{1}{n^3 + n}$ is convergent using Comparison.
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Converges; compare to $\frac{1}{n^3}$ (convergent $p$-series). For large $n$, $\frac{1}{n^3 + n} \sim \frac{1}{n^3}$.
Converges; compare to $\frac{1}{n^3}$ (convergent $p$-series). For large $n$, $\frac{1}{n^3 + n} \sim \frac{1}{n^3}$.
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Identify a series to compare with $\frac{2n^2+1}{n^3+3}$.
Identify a series to compare with $\frac{2n^2+1}{n^3+3}$.
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Compare with $\frac{1}{n}$, a divergent $p$-series. For large $n$, $\frac{2n^2+1}{n^3+3} \sim \frac{2}{n}$.
Compare with $\frac{1}{n}$, a divergent $p$-series. For large $n$, $\frac{2n^2+1}{n^3+3} \sim \frac{2}{n}$.
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Identify if $\frac{1}{n^2}$ converges using the Comparison Test.
Identify if $\frac{1}{n^2}$ converges using the Comparison Test.
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Converges; compare to $\frac{1}{n^2}$ (convergent $p$-series). $\frac{1}{n^2}$ is itself a convergent $p$-series with $p = 2 > 1$.
Converges; compare to $\frac{1}{n^2}$ (convergent $p$-series). $\frac{1}{n^2}$ is itself a convergent $p$-series with $p = 2 > 1$.
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What is the Comparison Test for convergence?
What is the Comparison Test for convergence?
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A test comparing a series to a known convergent or divergent series. Direct comparison determines convergence by relating to known series.
A test comparing a series to a known convergent or divergent series. Direct comparison determines convergence by relating to known series.
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Use Comparison: does $\frac{1}{n^{1.1}}$ converge?
Use Comparison: does $\frac{1}{n^{1.1}}$ converge?
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Converges; compare to $\frac{1}{n^{1.1}}$ (convergent $p$-series). $p = 1.1 > 1$ makes this a convergent $p$-series.
Converges; compare to $\frac{1}{n^{1.1}}$ (convergent $p$-series). $p = 1.1 > 1$ makes this a convergent $p$-series.
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Which series should you choose for the Comparison Test?
Which series should you choose for the Comparison Test?
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Choose a series $b_n$ that is similar in form to $a_n$. Similar form makes the comparison meaningful and easier to evaluate.
Choose a series $b_n$ that is similar in form to $a_n$. Similar form makes the comparison meaningful and easier to evaluate.
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