If limn→∞an=0 and an is decreasing, then the alternating series ∑(−1)nan converges.
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Explain the Alternating Series Test.
If limn→∞an=0 and an is decreasing, then the alternating series ∑(−1)nan converges.
Why is it important that an decreases in the Alternating Series Test?
It ensures that the terms are getting smaller in magnitude, allowing the partial sums to converge.
What happens if limn→∞an=0 in an alternating series?
The series diverges by the Divergence Test.
Does the Alternating Series Test determine absolute convergence?
No, it only determines conditional convergence. It doesn't tell us if ∑∣an∣ converges.
What is the significance of cos(nπ) in alternating series?
cos(nπ) is equivalent to (−1)n, providing the alternating sign for the series.
State the Alternating Series Test.
If an>0, limn→∞an=0, and an is a decreasing sequence, then the alternating series ∑n=1∞(−1)nan converges.
How to test ∑n=1∞n(−1)n for convergence?
Identify an=n1. 2. Check limn→∞n1=0. 3. Verify n1>n+11. Since both conditions are met, the series converges.
How to test ∑n=1∞n+1(−1)nn for convergence?
Identify an=n+1n. 2. Check limn→∞n+1n=1=0. Since the limit is not 0, the series diverges.
How to test ∑n=1∞ln(n)(−1)n for convergence?
Identify an=ln(n)1. 2. Check limn→∞ln(n)1=0. 3. Verify ln(n)1>ln(n+1)1. Since both conditions are met, the series converges.
How to test ∑n=1∞2n(−1)nn2 for convergence?
Identify an=2nn2. 2. Check limn→∞2nn2=0 (using L'Hopital's rule). 3. Verify 2nn2>2n+1(n+1)2 (true for large n). Since both conditions are met, the series converges.