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  1. AP Calculus
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How to determine convergence/divergence using the nth Term Test?

  1. Find lim⁡n→∞an\lim_{n \to \infty} a_nlimn→∞​an​. 2. If the limit is not 0, the series diverges. 3. If the limit is 0, the test is inconclusive.
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How to determine convergence/divergence using the nth Term Test?

  1. Find lim⁡n→∞an\lim_{n \to \infty} a_nlimn→∞​an​. 2. If the limit is not 0, the series diverges. 3. If the limit is 0, the test is inconclusive.

How to apply the Limit Comparison Test?

  1. Choose a series ∑bn\sum b_n∑bn​ to compare with. 2. Find lim⁡n→∞anbn=c\lim_{n \to \infty} \frac{a_n}{b_n} = climn→∞​bn​an​​=c. 3. If ccc is finite and positive, both series converge or diverge together.

How to apply the Direct Comparison Test?

  1. Find a series to compare. 2. Establish inequality. 3. If larger converges, smaller converges. If smaller diverges, larger diverges.

How to apply the Integral Test?

  1. Verify f(x)f(x)f(x) is continuous, positive, decreasing. 2. Evaluate ∫1∞f(x)dx\int_{1}^{\infty} f(x) dx∫1∞​f(x)dx. 3. If the integral converges, the series converges. If the integral diverges, the series diverges.

How to apply the Alternating Series Test?

  1. Check if terms decrease in absolute value. 2. Check if lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0limn→∞​an​=0. 3. If both conditions are met, the series converges.

How to apply the Ratio Test?

  1. Find lim⁡n→∞∣an+1an∣=L\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = Llimn→∞​∣an​an+1​​∣=L. 2. If L<1L < 1L<1, the series converges. 3. If L>1L > 1L>1, the series diverges. 4. If L=1L = 1L=1, the test is inconclusive.

How to find the radius of convergence of a power series?

  1. Use the Ratio Test. 2. Solve for ∣x−c∣<R|x - c| < R∣x−c∣<R. 3. RRR is the radius of convergence.

How to find the interval of convergence of a power series?

  1. Find the radius of convergence, R. 2. Test the endpoints c−Rc - Rc−R and c+Rc + Rc+R for convergence. 3. Write the interval, including or excluding endpoints based on convergence.

How to estimate the sum of an alternating series with a specified error?

  1. Use Alternating Series Error Bound: ∣Error∣≤∣an+1∣|Error| \leq |a_{n+1}|∣Error∣≤∣an+1​∣. 2. Find the smallest nnn such that ∣an+1∣|a_{n+1}|∣an+1​∣ is less than the specified error. 3. Sum the first nnn terms.

How to find a Taylor series for a given function?

  1. Find derivatives of the function. 2. Evaluate derivatives at the center. 3. Plug into the Taylor series formula: ∑n=0∞f(n)(c)n!(x−c)n\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n∑n=0∞​n!f(n)(c)​(x−c)n.

What does the Alternating Series Test state?

If ana_nan​ is decreasing and lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0limn→∞​an​=0, then the alternating series ∑(−1)nan\sum (-1)^n a_n∑(−1)nan​ converges.

What does the Ratio Test state?

If lim⁡n→∞∣an+1an∣=L\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = Llimn→∞​∣an​an+1​​∣=L, then the series converges if L<1L < 1L<1, diverges if L>1L > 1L>1, and is inconclusive if L=1L = 1L=1.

What does the nth Term Test for Divergence state?

If lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0limn→∞​an​=0, then the series ∑an\sum a_n∑an​ diverges.

What is the formula for the nth term of an arithmetic sequence?

a+(n−1)da + (n-1)da+(n−1)d, where aaa is the first term and ddd is the common difference.

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

a∗rn−1a * r^{n-1}a∗rn−1, where aaa is the first term and rrr is the common ratio.

What is the general form of a power series?

∑an(x−c)n\sum a_n(x-c)^n∑an​(x−c)n

What is the Maclaurin series for exe^xex?

∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!}∑n=0∞​n!xn​

What is the Maclaurin series for sin⁡(x)\sin(x)sin(x)?

∑n=0∞(−1)nx2n+1(2n+1)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}∑n=0∞​(2n+1)!(−1)nx2n+1​

What is the Maclaurin series for cos⁡(x)\cos(x)cos(x)?

∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}∑n=0∞​(2n)!(−1)nx2n​

What is the formula for Lagrange Error Bound?

∣Rn(x)∣≤M(n+1)!∣x−c∣n+1|R_n(x)| \leq \frac{M}{(n+1)!}|x-c|^{n+1}∣Rn​(x)∣≤(n+1)!M​∣x−c∣n+1, where M is the maximum value of the (n+1)th derivative.

What is the formula for Alternating Series Error Bound?

∣Error∣≤∣an+1∣|Error| \leq |a_{n+1}|∣Error∣≤∣an+1​∣, where an+1a_{n+1}an+1​ is the (n+1)th term of the series.