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  2. AP Calculus
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What is the general form of a p-series?

∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p}∑n=1∞​np1​

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What is the general form of a p-series?

∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p}∑n=1∞​np1​

What is the general form of a geometric series?

∑n=0∞arn\sum_{n=0}^{\infty} ar^n∑n=0∞​arn

Direct Comparison Test: If 0leanlebn0 le a_n le b_n0lean​lebn​ and ∑bn\sum b_n∑bn​ converges, what can you conclude?

∑an\sum a_n∑an​ converges.

Direct Comparison Test: If 0leanlebn0 le a_n le b_n0lean​lebn​ and ∑an\sum a_n∑an​ diverges, what can you conclude?

∑bn\sum b_n∑bn​ diverges.

Limit Comparison Test: What is the limit to evaluate?

lim⁡n→∞anbn\lim_{n\to\infty} \frac{a_n}{b_n}limn→∞​bn​an​​

Limit Comparison Test: What conclusion can be drawn if lim⁡n→∞anbn=c\lim_{n\to\infty} \frac{a_n}{b_n} = climn→∞​bn​an​​=c, where 0<c<∞0 < c < \infty0<c<∞?

Both ∑an\sum a_n∑an​ and ∑bn\sum b_n∑bn​ either converge or diverge.

What is the formula for the nth term of a geometric series?

arn−1ar^{n-1}arn−1

When does a p-series converge?

When p>1p > 1p>1

When does a p-series diverge?

When p≤1p \le 1p≤1

When does a geometric series converge?

When ∣r∣<1|r| < 1∣r∣<1

What does the Direct Comparison Theorem state?

For series ∑an\sum a_n∑an​ and ∑bn\sum b_n∑bn​ with 0≤an≤bn0 \le a_n \le b_n0≤an​≤bn​, if ∑bn\sum b_n∑bn​ converges, then ∑an\sum a_n∑an​ converges. If ∑an\sum a_n∑an​ diverges, then ∑bn\sum b_n∑bn​ diverges.

What does the Limit Comparison Theorem state?

For series ∑an\sum a_n∑an​ and ∑bn\sum b_n∑bn​ with an,bn>0a_n, b_n > 0an​,bn​>0, if lim⁡n→∞anbn=c\lim_{n\to\infty} \frac{a_n}{b_n} = climn→∞​bn​an​​=c, where 0<c<∞0 < c < \infty0<c<∞, then both series either converge or diverge.

What does the p-series Test theorem state?

The p-series ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p}∑n=1∞​np1​ converges if p>1p > 1p>1 and diverges if p≤1p \le 1p≤1.

What does the Geometric Series Test theorem state?

The geometric series ∑n=0∞arn\sum_{n=0}^{\infty} ar^n∑n=0∞​arn converges if ∣r∣<1|r| < 1∣r∣<1 and diverges if ∣r∣≥1|r| \ge 1∣r∣≥1.

How is the Direct Comparison Theorem applied?

Find a series ∑bn\sum b_n∑bn​ whose convergence/divergence is known, and show that 0≤an≤bn0 \le a_n \le b_n0≤an​≤bn​ (for convergence) or an≥bna_n \ge b_nan​≥bn​ (for divergence).

How is the Limit Comparison Theorem applied?

Find a series ∑bn\sum b_n∑bn​ and compute lim⁡n→∞anbn\lim_{n\to\infty} \frac{a_n}{b_n}limn→∞​bn​an​​. If the limit is finite and positive, the series behave alike.

What are the limitations of the Direct Comparison Theorem?

It requires finding a suitable inequality, which can be difficult. It's inconclusive if the inequality goes the wrong way.

What are the limitations of the Limit Comparison Theorem?

The limit must be finite and positive. If the limit is 0 or infinity, the test is inconclusive.

What is the role of the condition an,bn≥0a_n, b_n \ge 0an​,bn​≥0 in the Comparison Theorems?

It ensures that the inequalities used in the theorems are valid. If terms are negative, the comparison may not hold.

How does L'Hopital's Rule relate to the Limit Comparison Theorem?

L'Hopital's Rule can be used to evaluate the limit lim⁡n→∞anbn\lim_{n\to\infty} \frac{a_n}{b_n}limn→∞​bn​an​​ when it results in an indeterminate form.

What are the differences between the Direct Comparison Test and the Limit Comparison Test?

Direct Comparison: Requires direct inequality (an≤bna_n \le b_nan​≤bn​). Limit Comparison: Compares the limit of the ratio of terms. Direct Comparison: More straightforward when applicable. Limit Comparison: Useful when direct inequality is hard to establish.

Compare and contrast p-series and geometric series.

p-series: Form ∑1np\sum \frac{1}{n^p}∑np1​, converges if p>1p > 1p>1. Geometric series: Form ∑arn\sum ar^n∑arn, converges if ∣r∣<1|r| < 1∣r∣<1. Both: Useful for comparison tests. p-series: Depends on exponent p. Geometric series: Depends on common ratio r.

Compare Direct Comparison Test when an<bna_n < b_nan​<bn​ and ∑bn\sum b_n∑bn​ converges vs. when ∑an\sum a_n∑an​ diverges.

∑bn\sum b_n∑bn​ converges: Implies ∑an\sum a_n∑an​ converges. ∑an\sum a_n∑an​ diverges: Implies ∑bn\sum b_n∑bn​ diverges. The first shows convergence, the second shows divergence.

Compare Limit Comparison Test when the limit is a finite positive number vs. when the limit is zero or infinity.

Finite positive number: Both series behave alike (both converge or both diverge). Zero or infinity: The test is inconclusive and a different comparison series must be found.

Compare using bn=1npb_n = \frac{1}{n^p}bn​=np1​ with p>1p>1p>1 and p≤1p \le 1p≤1 in comparison tests.

p>1p>1p>1: ∑bn\sum b_n∑bn​ converges, useful for showing convergence of a smaller series. p≤1p \le 1p≤1: ∑bn\sum b_n∑bn​ diverges, useful for showing divergence of a larger series.

What are the similarities and differences between Direct Comparison Test and Integral Test?

Direct Comparison: Compares series to another series. Integral Test: Relates series convergence to integral convergence. Both: Used to determine convergence/divergence. Integral Test: Requires function to be continuous, positive, and decreasing.

Compare the conditions for convergence of a geometric series with r=0.5 and r=2.

r=0.5: Converges because |0.5| < 1. r=2: Diverges because |2| > 1. Convergence depends on the absolute value of r being less than 1.

Compare using Direct Comparison with ana_nan​ and bnb_nbn​ where ana_nan​ is always greater than bnb_nbn​.

If an>bna_n > b_nan​>bn​ and ∑an\sum a_n∑an​ converges, the test is inconclusive. If an>bna_n > b_nan​>bn​ and ∑bn\sum b_n∑bn​ diverges, then ∑an\sum a_n∑an​ diverges.

Compare the usefulness of comparison tests for series with alternating signs vs. series with only positive terms.

Comparison tests are typically used for series with positive terms. Alternating series often require the Alternating Series Test.

Compare the difficulty of applying the Direct Comparison Test when the inequality is obvious vs. when it requires manipulation.

Obvious inequality: Application is straightforward. Requires manipulation: More complex, requires algebraic skills to establish the inequality.