zuai-logo

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

a+(n1)da + (n-1)d, where aa is the first term and dd is the common difference.

Flip to see [answer/question]
Flip to see [answer/question]

All Flashcards

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

a+(n1)da + (n-1)d, where aa is the first term and dd is the common difference.

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

arn1a * r^{n-1}, where aa is the first term and rr is the common ratio.

What is the general form of a power series?

an(xc)n\sum a_n(x-c)^n

What is the Maclaurin series for exe^x?

n=0xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!}

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

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

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

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

What is the formula for Lagrange Error Bound?

Rn(x)M(n+1)!xcn+1|R_n(x)| \leq \frac{M}{(n+1)!}|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?

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

How to determine convergence/divergence using the nth Term Test?

  1. Find limnan\lim_{n \to \infty} a_n. 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 to compare with. 2. Find limnanbn=c\lim_{n \to \infty} \frac{a_n}{b_n} = c. 3. If cc 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) is continuous, positive, decreasing. 2. Evaluate 1f(x)dx\int_{1}^{\infty} 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 limnan=0\lim_{n \to \infty} a_n = 0. 3. If both conditions are met, the series converges.

How to apply the Ratio Test?

  1. Find limnan+1an=L\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L. 2. If L<1L < 1, the series converges. 3. If L>1L > 1, the series diverges. 4. If L=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 xc<R|x - c| < R. 3. RR 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 cRc - R and c+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: Erroran+1|Error| \leq |a_{n+1}|. 2. Find the smallest nn such that an+1|a_{n+1}| is less than the specified error. 3. Sum the first nn 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=0f(n)(c)n!(xc)n\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n.

Explain the nth Term Test for Divergence.

If limnan0\lim_{n \to \infty} a_n \neq 0, then the series an\sum a_n diverges.

Explain the Limit Comparison Test.

If limnanbn=c\lim_{n \to \infty} \frac{a_n}{b_n} = c, where cc is finite and positive, then an\sum a_n and bn\sum b_n either both converge or both diverge.

Explain the Direct Comparison Test.

If 0anbn0 \leq a_n \leq b_n and bn\sum b_n converges, then an\sum a_n converges. If anbn0a_n \geq b_n \geq 0 and bn\sum b_n diverges, then an\sum a_n diverges.

Explain the Integral Test.

If f(x)f(x) is continuous, positive, and decreasing for x1x \geq 1, then n=1f(n)\sum_{n=1}^{\infty} f(n) and 1f(x)dx\int_{1}^{\infty} f(x) dx either both converge or both diverge.

Explain the Alternating Series Test.

If ana_n is decreasing and limnan=0\lim_{n \to \infty} a_n = 0, then the alternating series (1)nan\sum (-1)^n a_n converges.

Explain the Ratio Test.

If limnan+1an=L\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L, then the series converges if L<1L < 1, diverges if L>1L > 1, and is inconclusive if L=1L = 1.

Explain the concept of radius of convergence.

The distance from the center of a power series within which the series converges.

Explain the Alternating Series Error Bound.

The error in approximating an alternating series is less than or equal to the absolute value of the first omitted term.

Explain the Lagrange Error Bound.

The error in approximating a function using a Taylor polynomial is bounded by the Lagrange Error Bound, which depends on the (n+1)th derivative.

What does it mean for a series to converge absolutely?

A series an\sum a_n converges absolutely if an\sum |a_n| converges.

What does it mean for a series to converge conditionally?

A series an\sum a_n converges conditionally if an\sum a_n converges but an\sum |a_n| diverges.