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Questions and Answers
If a series ∑an converges, which of the following statements is true about the sequence of terms {an}?
If a series ∑an converges, which of the following statements is true about the sequence of terms {an}?
- The sequence of terms {an} diverges to ∞.
- The sequence of terms {an} converges to 0. (correct)
- The sequence of terms {an} converges to 1.
- The sequence of terms {an} oscillates.
What is the sum of a convergent geometric series ∑ar^(n-1)?
What is the sum of a convergent geometric series ∑ar^(n-1)?
- a/(1 - r) (correct)
- a*r
- a/(1 + r)
- a/r
If a p-series ∑(1/n^p) converges, what can be said about the value of p?
If a p-series ∑(1/n^p) converges, what can be said about the value of p?
- p < 1
- p = 1
- p ≤ 0
- p > 1 (correct)
If 0 ≤ an ≤ bn for all n, and ∑bn diverges, what can be said about the convergence of ∑an?
If 0 ≤ an ≤ bn for all n, and ∑bn diverges, what can be said about the convergence of ∑an?
If lim (an / bn) = L, where 0 < L < ∞, what can be said about the convergence of ∑an and ∑bn?
If lim (an / bn) = L, where 0 < L < ∞, what can be said about the convergence of ∑an and ∑bn?
If lim (|an+1 / an|) = L, where L = 1, what can be said about the convergence of the series ∑an?
If lim (|an+1 / an|) = L, where L = 1, what can be said about the convergence of the series ∑an?
If lim (|an|^(1/n)) = L, where L = 1, what can be said about the convergence of the series ∑an?
If lim (|an|^(1/n)) = L, where L = 1, what can be said about the convergence of the series ∑an?
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Study Notes
Convergence Tests for Series of Real Numbers
Nth Term Test
- If the sequence of terms {an} converges to 0, then the series ∑an converges.
- If the sequence of terms {an} does not converge to 0, then the series ∑an diverges.
Geometric Series Test
- A geometric series ∑ar^(n-1) converges if |r| < 1 and diverges if |r| ≥ 1.
- The sum of a convergent geometric series is a/(1 - r).
p-Series Test
- A p-series ∑(1/n^p) converges if p > 1 and diverges if p ≤ 1.
Comparison Test
- If 0 ≤ an ≤ bn for all n, and ∑bn converges, then ∑an converges.
- If an ≥ bn for all n, and ∑bn diverges, then ∑an diverges.
Limit Comparison Test
- If lim (an / bn) = L, where 0 < L < ∞, then ∑an and ∑bn both converge or both diverge.
Ratio Test
- If lim (|an+1 / an|) = L, then:
- If L < 1, the series ∑an converges.
- If L > 1, the series ∑an diverges.
- If L = 1, the test is inconclusive.
Root Test
- If lim (|an|^(1/n)) = L, then:
- If L < 1, the series ∑an converges.
- If L > 1, the series ∑an diverges.
- If L = 1, the test is inconclusive.
These convergence tests can be used to determine whether a series of real numbers converges or diverges.
Convergence Tests for Series of Real Numbers
Nth Term Test
- Necessary condition for series convergence: sequence of terms {an} converges to 0
- Sufficient condition for series divergence: sequence of terms {an} does not converge to 0
Geometric Series Test
- Convergence condition: |r| < 1
- Divergence condition: |r| ≥ 1
- Sum of a convergent geometric series: a/(1 - r)
p-Series Test
- Convergence condition: p > 1
- Divergence condition: p ≤ 1
Comparison Test
- Convergence condition: 0 ≤ an ≤ bn for all n, and ∑bn converges
- Divergence condition: an ≥ bn for all n, and ∑bn diverges
Limit Comparison Test
- Convergence/divergence condition: lim (an / bn) = L, where 0 < L < ∞
Ratio Test
- Convergence condition: lim (|an+1 / an|) < 1
- Divergence condition: lim (|an+1 / an|) > 1
- Inconclusive condition: lim (|an+1 / an|) = 1
Root Test
- Convergence condition: lim (|an|^(1/n)) < 1
- Divergence condition: lim (|an|^(1/n)) > 1
- Inconclusive condition: lim (|an|^(1/n)) = 1
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