How to determine convergence/divergence using the nth Term Test?
1. Find $\lim_{n \to \infty} a_n$. 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 $\sum b_n$ to compare with. 2. Find $\lim_{n \to \infty} \frac{a_n}{b_n} = c$. 3. If $c$ 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 $f(x)$ is continuous, positive, decreasing. 2. Evaluate $\int_{1}^{\infty} f(x) dx$. 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 $\lim_{n \to \infty} a_n = 0$. 3. If both conditions are met, the series converges.
How to apply the Ratio Test?
1. Find $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$. 2. If $L < 1$, the series converges. 3. If $L > 1$, the series diverges. 4. If $L = 1$, the test is inconclusive.
How to find the radius of convergence of a power series?
1. Use the Ratio Test. 2. Solve for $|x - c| < R$. 3. $R$ 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 $c - R$ and $c + R$ 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: $|Error| \leq |a_{n+1}|$. 2. Find the smallest $n$ such that $|a_{n+1}|$ is less than the specified error. 3. Sum the first $n$ 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: $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$.
What is the formula for the nth term of an arithmetic sequence?
$a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
What is the formula for the nth term of a geometric sequence?
$a * r^{n-1}$, where $a$ is the first term and $r$ is the common ratio.
What is the general form of a power series?
$\sum a_n(x-c)^n$
What is the Maclaurin series for $e^x$?
$\sum_{n=0}^{\infty} \frac{x^n}{n!}$
What is the Maclaurin series for $\sin(x)$?
$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$
What is the Maclaurin series for $\cos(x)$?
$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$
What is the formula for Lagrange Error Bound?
$|R_n(x)| \leq \frac{M}{(n+1)!}|x-c|^{n+1}$, where M is the maximum value of the (n+1)th derivative.
What is the formula for Alternating Series Error Bound?
$|Error| \leq |a_{n+1}|$, where $a_{n+1}$ is the (n+1)th term of the series.
Explain the nth Term Test for Divergence.
If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.
Explain the Limit Comparison Test.
If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $c$ is finite and positive, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.
Explain the Direct Comparison Test.
If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $a_n \geq b_n \geq 0$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.
Explain the Integral Test.
If $f(x)$ is continuous, positive, and decreasing for $x \geq 1$, then $\sum_{n=1}^{\infty} f(n)$ and $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.
Explain the Alternating Series Test.
If $a_n$ is decreasing and $\lim_{n \to \infty} a_n = 0$, then the alternating series $\sum (-1)^n a_n$ converges.
Explain the Ratio Test.
If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$, then the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.
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 $\sum a_n$ converges absolutely if $\sum |a_n|$ converges.
What does it mean for a series to converge conditionally?
A series $\sum a_n$ converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges.
