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  1. AP Calculus
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Define a sequence.

A function whose domain is the set of natural numbers.

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Define a sequence.

A function whose domain is the set of natural numbers.

Define a series.

The sum of the terms of a sequence.

What is a convergent sequence/series?

Terms approach a specific value (the limit).

What is a divergent sequence/series?

Terms do not approach a specific value; the sum is infinite.

Define an arithmetic sequence.

Sequence with a constant difference between consecutive terms.

Define a geometric sequence.

Sequence where each term is the previous term multiplied by a constant ratio.

Define a harmonic series.

Series where terms are reciprocals of positive integers.

Define a power series.

Series of the form ∑an(x−c)n\sum a_n(x-c)^n∑an​(x−c)n where ana_nan​ are constants and ccc is a constant.

Define an alternating series.

Series where the terms alternate in sign.

Define a Taylor series.

Representation of a function as an infinite sum of terms, with each term being a polynomial function of a single variable.

Define a Maclaurin series.

A Taylor series centered at 0.

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.

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.