Harmonic Series and p-Series - AP Calculus BC
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What condition makes a p-series diverge?
What condition makes a p-series diverge?
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Diverges if $p \leq 1$. When $p \leq 1$, terms don't decrease fast enough.
Diverges if $p \leq 1$. When $p \leq 1$, terms don't decrease fast enough.
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Does the series $\sum_{n=1}^{\infty} \frac{1}{n}$ converge or diverge?
Does the series $\sum_{n=1}^{\infty} \frac{1}{n}$ converge or diverge?
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Diverges. The harmonic series is the classic divergent series.
Diverges. The harmonic series is the classic divergent series.
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State the formula for a p-series.
State the formula for a p-series.
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$\sum_{n=1}^{\infty} \frac{1}{n^p}$. General form where $p$ determines convergence behavior.
$\sum_{n=1}^{\infty} \frac{1}{n^p}$. General form where $p$ determines convergence behavior.
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Does $\sum_{n=1}^{\infty} \frac{1}{n^{0.7}}$ converge or diverge?
Does $\sum_{n=1}^{\infty} \frac{1}{n^{0.7}}$ converge or diverge?
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Diverges. Since $p = 0.7 < 1$, this p-series diverges.
Diverges. Since $p = 0.7 < 1$, this p-series diverges.
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Is $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$ convergent or divergent?
Is $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$ convergent or divergent?
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Divergent. Since $p = 0.5 \leq 1$, this p-series diverges.
Divergent. Since $p = 0.5 \leq 1$, this p-series diverges.
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Identify the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^2}$.
Identify the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^2}$.
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Converges. Since $p = 2 > 1$, this p-series converges.
Converges. Since $p = 2 > 1$, this p-series converges.
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What is the nature of $\sum_{n=1}^{\infty} \frac{1}{n^{2.1}}$?
What is the nature of $\sum_{n=1}^{\infty} \frac{1}{n^{2.1}}$?
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Convergent. Since $p = 2.1 > 1$, this p-series converges.
Convergent. Since $p = 2.1 > 1$, this p-series converges.
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What is the definition of a harmonic series?
What is the definition of a harmonic series?
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The series $\sum_{n=1}^{\infty} \frac{1}{n}$. The classic divergent series with terms $\frac{1}{n}$.
The series $\sum_{n=1}^{\infty} \frac{1}{n}$. The classic divergent series with terms $\frac{1}{n}$.
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Identify if the series $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$ converges.
Identify if the series $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$ converges.
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Converges. Since $p = 1.5 > 1$, this p-series converges.
Converges. Since $p = 1.5 > 1$, this p-series converges.
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What condition makes a p-series converge?
What condition makes a p-series converge?
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Converges if $p > 1$. When $p > 1$, the terms decrease fast enough for convergence.
Converges if $p > 1$. When $p > 1$, the terms decrease fast enough for convergence.
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Does the series $\sum_{n=1}^{\infty} \frac{1}{n^{3.5}}$ converge?
Does the series $\sum_{n=1}^{\infty} \frac{1}{n^{3.5}}$ converge?
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Converges. Since $p = 3.5 > 1$, this p-series converges.
Converges. Since $p = 3.5 > 1$, this p-series converges.
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Identify the type of series: $\sum_{n=1}^{\infty} \frac{1}{n^{0.9}}$.
Identify the type of series: $\sum_{n=1}^{\infty} \frac{1}{n^{0.9}}$.
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p-series. Has form $\sum \frac{1}{n^p}$ with $p = 0.9$.
p-series. Has form $\sum \frac{1}{n^p}$ with $p = 0.9$.
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State the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^{0.8}}$.
State the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^{0.8}}$.
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Diverges. Since $p = 0.8 < 1$, this p-series diverges.
Diverges. Since $p = 0.8 < 1$, this p-series diverges.
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Determine the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^{4}}$.
Determine the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^{4}}$.
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Converges. Since $p = 4 > 1$, this p-series converges.
Converges. Since $p = 4 > 1$, this p-series converges.
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What is the second term of a harmonic series?
What is the second term of a harmonic series?
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$\frac{1}{2}$. The second term in the harmonic series is $\frac{1}{2}$.
$\frac{1}{2}$. The second term in the harmonic series is $\frac{1}{2}$.
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Identify if $\sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$ converges or diverges.
Identify if $\sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$ converges or diverges.
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Diverges. Since $p = 0.5 < 1$, this p-series diverges.
Diverges. Since $p = 0.5 < 1$, this p-series diverges.
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Identify if the series $\sum_{n=1}^{\infty} \frac{1}{n^{0.6}}$ is convergent.
Identify if the series $\sum_{n=1}^{\infty} \frac{1}{n^{0.6}}$ is convergent.
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Diverges. Since $p = 0.6 < 1$, this p-series diverges.
Diverges. Since $p = 0.6 < 1$, this p-series diverges.
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Identify if $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ converges or diverges.
Identify if $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ converges or diverges.
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Converges. Since $p = 1.5 > 1$, this p-series converges.
Converges. Since $p = 1.5 > 1$, this p-series converges.
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Identify the series that converges: $\sum_{n=1}^{\infty} \frac{1}{n^3}$ or $\sum_{n=1}^{\infty} \frac{1}{n}$.
Identify the series that converges: $\sum_{n=1}^{\infty} \frac{1}{n^3}$ or $\sum_{n=1}^{\infty} \frac{1}{n}$.
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$\sum_{n=1}^{\infty} \frac{1}{n^3}$. Since $p = 3 > 1$, the first series converges.
$\sum_{n=1}^{\infty} \frac{1}{n^3}$. Since $p = 3 > 1$, the first series converges.
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Which series is divergent: $\sum_{n=1}^{\infty} \frac{1}{n}$ or $\sum_{n=1}^{\infty} \frac{1}{n^2}$?
Which series is divergent: $\sum_{n=1}^{\infty} \frac{1}{n}$ or $\sum_{n=1}^{\infty} \frac{1}{n^2}$?
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$\sum_{n=1}^{\infty} \frac{1}{n}$. The harmonic series diverges while $\sum \frac{1}{n^2}$ converges.
$\sum_{n=1}^{\infty} \frac{1}{n}$. The harmonic series diverges while $\sum \frac{1}{n^2}$ converges.
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State whether $\sum_{n=1}^{\infty} \frac{1}{n^{2.5}}$ converges.
State whether $\sum_{n=1}^{\infty} \frac{1}{n^{2.5}}$ converges.
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Converges. Since $p = 2.5 > 1$, this p-series converges.
Converges. Since $p = 2.5 > 1$, this p-series converges.
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For which $p$ is the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ divergent?
For which $p$ is the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ divergent?
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$p \leq 1$. P-series diverge when $p$ is at most 1.
$p \leq 1$. P-series diverge when $p$ is at most 1.
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What is the fifth term of a harmonic series?
What is the fifth term of a harmonic series?
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$\frac{1}{5}$. The fifth term in the harmonic series is $\frac{1}{5}$.
$\frac{1}{5}$. The fifth term in the harmonic series is $\frac{1}{5}$.
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What is the fourth term of a harmonic series?
What is the fourth term of a harmonic series?
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$\frac{1}{4}$. The fourth term in the harmonic series is $\frac{1}{4}$.
$\frac{1}{4}$. The fourth term in the harmonic series is $\frac{1}{4}$.
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What is the condition for the convergence of a harmonic series?
What is the condition for the convergence of a harmonic series?
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Always diverges. The harmonic series is famously always divergent.
Always diverges. The harmonic series is famously always divergent.
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What is the third term of a harmonic series?
What is the third term of a harmonic series?
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$\frac{1}{3}$. The third term in the harmonic series is $\frac{1}{3}$.
$\frac{1}{3}$. The third term in the harmonic series is $\frac{1}{3}$.
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What is the first term of a harmonic series?
What is the first term of a harmonic series?
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- The harmonic series starts with $\frac{1}{1} = 1$.
- The harmonic series starts with $\frac{1}{1} = 1$.
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What type of series is $\sum_{n=1}^{\infty} \frac{1}{n^{0.3}}$?
What type of series is $\sum_{n=1}^{\infty} \frac{1}{n^{0.3}}$?
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p-series. Has form $\sum \frac{1}{n^p}$ with $p = 0.3$.
p-series. Has form $\sum \frac{1}{n^p}$ with $p = 0.3$.
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What type of series is $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$?
What type of series is $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$?
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p-series. Has the form $\sum \frac{1}{n^p}$ with $p = 1.5$.
p-series. Has the form $\sum \frac{1}{n^p}$ with $p = 1.5$.
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What is the definition of a harmonic series?
What is the definition of a harmonic series?
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The series $\sum_{n=1}^{\infty} \frac{1}{n}$. The classic divergent series with terms $\frac{1}{n}$.
The series $\sum_{n=1}^{\infty} \frac{1}{n}$. The classic divergent series with terms $\frac{1}{n}$.
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