Determining Absolute or Conditional Convergence - AP Calculus BC
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Which test is most suitable for factorial expressions?
Which test is most suitable for factorial expressions?
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The Ratio Test. Effective for factorial terms.
The Ratio Test. Effective for factorial terms.
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What does the Root Test involve calculating?
What does the Root Test involve calculating?
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$\lim_{n \to \infty} \sqrt[n]{|a_n|}$. Formula for root test calculation.
$\lim_{n \to \infty} \sqrt[n]{|a_n|}$. Formula for root test calculation.
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State the condition for a series to converge using the Alternating Series Test.
State the condition for a series to converge using the Alternating Series Test.
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$a_n$ decreases to $0$ and $a_n > a_{n+1}$. Terms must decrease monotonically to zero.
$a_n$ decreases to $0$ and $a_n > a_{n+1}$. Terms must decrease monotonically to zero.
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What is the first step in the Alternating Series Test?
What is the first step in the Alternating Series Test?
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Check if $a_n$ is decreasing. Monotonic decreasing requirement.
Check if $a_n$ is decreasing. Monotonic decreasing requirement.
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Determine convergence: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.
Determine convergence: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.
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Converges conditionally. Passes Alternating Series Test but absolute values diverge.
Converges conditionally. Passes Alternating Series Test but absolute values diverge.
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Apply the Ratio Test: $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$.
Apply the Ratio Test: $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$.
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Converges absolutely. Factorials grow faster than exponentials.
Converges absolutely. Factorials grow faster than exponentials.
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Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n}$.
Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n}$.
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Diverges. P-series with $p=1$ diverges.
Diverges. P-series with $p=1$ diverges.
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Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n$.
Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n$.
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Converges absolutely. Geometric series with ratio $1/3 < 1$.
Converges absolutely. Geometric series with ratio $1/3 < 1$.
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Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$.
Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$.
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Converges absolutely. Alternating p-series with $p=2>1$.
Converges absolutely. Alternating p-series with $p=2>1$.
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What test determines absolute convergence using absolute values?
What test determines absolute convergence using absolute values?
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The Absolute Convergence Test. Tests convergence of $\sum |a_n|$.
The Absolute Convergence Test. Tests convergence of $\sum |a_n|$.
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What characterizes a geometric series?
What characterizes a geometric series?
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Each term is a constant multiple of the previous term. Common ratio between consecutive terms.
Each term is a constant multiple of the previous term. Common ratio between consecutive terms.
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Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$.
Determine convergence: $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$.
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Converges conditionally. Alternating p-series with $p=1/2<1$.
Converges conditionally. Alternating p-series with $p=1/2<1$.
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What is the definition of absolute convergence?
What is the definition of absolute convergence?
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A series converges absolutely if the series of absolute values converges. Convergence when taking absolute values.
A series converges absolutely if the series of absolute values converges. Convergence when taking absolute values.
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What is the Ratio Test's condition for absolute convergence?
What is the Ratio Test's condition for absolute convergence?
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If $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$, converges absolutely. Ratio of consecutive terms approaches value less than 1.
If $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$, converges absolutely. Ratio of consecutive terms approaches value less than 1.
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What is the p-series test condition for convergence?
What is the p-series test condition for convergence?
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If $p > 1$, the p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges. Exponent determines convergence behavior.
If $p > 1$, the p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges. Exponent determines convergence behavior.
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What test can determine if a series converges conditionally?
What test can determine if a series converges conditionally?
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The Alternating Series Test. For series with alternating signs.
The Alternating Series Test. For series with alternating signs.
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Apply the Comparison Test: $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$.
Apply the Comparison Test: $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$.
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Converges absolutely. Compare with convergent p-series $\sum \frac{1}{n^2}$.
Converges absolutely. Compare with convergent p-series $\sum \frac{1}{n^2}$.
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What is the series $\sum_{n=1}^{\infty} \frac{1}{n}$ known as?
What is the series $\sum_{n=1}^{\infty} \frac{1}{n}$ known as?
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Harmonic Series. Famous divergent series.
Harmonic Series. Famous divergent series.
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Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^2}$.
Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^2}$.
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Converges absolutely. P-series with $p=2>1$ converges.
Converges absolutely. P-series with $p=2>1$ converges.
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Determine convergence: $\sum_{n=1}^{\infty} n^2 e^{-n}$.
Determine convergence: $\sum_{n=1}^{\infty} n^2 e^{-n}$.
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Converges absolutely. Exponential decay dominates polynomial growth.
Converges absolutely. Exponential decay dominates polynomial growth.
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What is required for the Limit Comparison Test?
What is required for the Limit Comparison Test?
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A second series $b_n$ to compare with $a_n$. Need comparison series with known behavior.
A second series $b_n$ to compare with $a_n$. Need comparison series with known behavior.
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Identify the series: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.
Identify the series: $\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.
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Geometric Series. Series with constant ratio between terms.
Geometric Series. Series with constant ratio between terms.
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Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{n}{n+1}\right)^n$.
Apply the Root Test: $\sum_{n=1}^{\infty} \left(\frac{n}{n+1}\right)^n$.
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Diverges. Root approaches 1, so diverges.
Diverges. Root approaches 1, so diverges.
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What is the condition for convergence using the Root Test?
What is the condition for convergence using the Root Test?
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If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, converges absolutely. nth root of terms approaches value less than 1.
If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, converges absolutely. nth root of terms approaches value less than 1.
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Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$.
Determine convergence: $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$.
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Diverges. P-series with $p=0.5<1$ diverges.
Diverges. P-series with $p=0.5<1$ diverges.
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Which test is most suitable for factorial expressions?
Which test is most suitable for factorial expressions?
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The Ratio Test. Effective for factorial terms.
The Ratio Test. Effective for factorial terms.
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Identify the series: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.
Identify the series: $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$.
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Alternating Harmonic Series. Classic conditionally convergent series.
Alternating Harmonic Series. Classic conditionally convergent series.
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Identify the limit for convergence in the Ratio Test.
Identify the limit for convergence in the Ratio Test.
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$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. Formula for ratio test calculation.
$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. Formula for ratio test calculation.
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What is the definition of conditional convergence?
What is the definition of conditional convergence?
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A series converges conditionally if it converges, but not absolutely. Series converges but not when absolute values are taken.
A series converges conditionally if it converges, but not absolutely. Series converges but not when absolute values are taken.
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What test compares series to determine absolute convergence?
What test compares series to determine absolute convergence?
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The Comparison Test. Compares term by term with known series.
The Comparison Test. Compares term by term with known series.
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