What is an infinite sequence in mathematics?
An ordered list of numbers indexed by positive integers or another integer set.
When does a sequence converge to a limit L?
When its terms become arbitrarily close to L as n approaches infinity.
What happens if a sequence has no finite limit?
The sequence diverges.
For which values of r does the sequence r^n converge?
For all r with -1 < r ≤ 1.
What is the limit of the sequence r^n when -1 < r < 1?
The limit is 0.
What is the limit of the sequence r^n when r = 1?
The limit is 1.
What does the Monotonic Sequence Theorem state?
Every bounded monotonic sequence converges.
Which monotonic sequences are guaranteed to converge?
Increasing sequences bounded above and decreasing sequences bounded below.
How is an infinite series defined?
By its partial sums.
When does an infinite series converge?
When its partial sums approach a finite real number.
What is the formula for the nth partial sum?
s_n = sum of a_i from i=1 to n.
When does the geometric series sum ar^(n-1) converge?
When the absolute value of r is less than 1.
What is the sum of a convergent geometric series?
a divided by (1 minus r).
What happens if the limit of a_n is nonzero or does not exist?
The series sum a_n diverges.
What if a_n approaches zero in the series sum a_n?
The test is inconclusive.
Does the harmonic series sum 1/n converge or diverge?
It diverges even though its terms approach zero.
What conditions on f ensure sum a_n and integral of f share convergence?
f must be continuous, positive, and decreasing on the relevant tail.
When does the p-series sum 1/n^p converge?
It converges when p is greater than 1.
When does the p-series sum 1/n^p diverge?
It diverges when p is less than or equal to 1.
What inequality bounds the remainder R_n for a positive decreasing series?
The remainder satisfies ∫_{n+1}^∞ f(x) dx ≤ R_n ≤ ∫_n^∞ f(x) dx.
What is the integral remainder bound for the series sum 1/n^3?
R_n is at most 1 divided by 2 n squared.
How many terms guarantee error below 0.0005 for sum 1/n^3 using integral remainder?
At least 32 terms guarantee the error is below 0.0005.
What happens if 0 ≤ a_n ≤ b_n and sum b_n converges?
Sum a_n converges.
What happens if a_n ≥ b_n ≥ 0 and sum b_n diverges?
Sum a_n diverges.
What does the limit comparison test say if limit of a_n/b_n is c with 0 < c < ∞?
Sum a_n and sum b_n both converge or both diverge.
How do you compare rational or algebraic terms to a p-series?
Compare dominant powers of n and choose bounds for convergence or divergence.
What inequality holds if 0 ≤ a_k ≤ b_k for every k > n?
Tail remainder R_n of sum a_k is at most tail T_n of sum b_k.
When does the alternating series sum (-1)^n b_n converge?
It converges when b_n is decreasing and approaches zero.
What inequality bounds the remainder of an alternating series meeting test conditions?
The remainder satisfies |R_n| ≤ b_{n+1}.
What defines absolute convergence of a series?
Absolute convergence occurs when the sum of |a_n| converges.
What defines conditional convergence of a series?
Conditional convergence occurs when sum a_n converges but sum |a_n| diverges.
What happens to the sum of an absolutely convergent series under rearrangement?
Its sum remains the same after any rearrangement.
How does rearranging a conditionally convergent series affect its sum?
Rearranging can change the sum of a conditionally convergent series.
What is the first step in classifying a series for convergence?
Inspect the term limit.
Which tests are suggested for factorials and constant-to-n terms?
The Ratio Test.
For which term form is the Root Test suggested?
Terms of the form (b_n)^n.
Why is the Ratio Test not recommended for p-series?
Because the ratio a_(n+1)/a_n tends to 1.
What is the geometric series template for |x|<1?
1/(1-x) equals the sum from n=0 to infinity of x^n.
How can new power-series representations be obtained?
By substituting an expression for x into a known series and adjusting convergence conditions.
How is the derivative of a power series inside its radius of convergence expressed?
The derivative equals the sum from n=1 to infinity of n c_n (x-a)^(n-1).
What happens to the radius of convergence after term-by-term differentiation or integration?
The radius of convergence remains the same.
What is the coefficient of \((x-a)^n\) in a Taylor series centered at a?
It is \(c_n=\frac{f^{(n)}(a)}{n!}\).
What is the formula for the nth Taylor polynomial centered at a?
It is \(T_n(x)=\sum_{i=0}^{n}\frac{f^{(i)}(a)}{i!}(x-a)^i\).
When does a formal Taylor series equal the function?
Only when the remainder \(R_n(x)\) approaches zero on the interval.
What does Taylor's Inequality state about the remainder \(R_n(x)\)?
If \(|f^{(n+1)}(x)| \le M\), then \(|R_n(x)| \le \frac{M|x-a|^{n+1}}{(n+1)!}\).
What is the Maclaurin series formula for e^x?
e^x = sum from n=0 to infinity of x^n divided by n!.
What is the radius of convergence for the Maclaurin series of e^x?
The radius of convergence is infinite.
What is the Maclaurin series formula for sin x?
sin x = sum from n=0 to infinity of (-1)^n x^(2n+1) divided by (2n+1)!.
What is the Maclaurin series formula for cos x?
cos x = sum from n=0 to infinity of (-1)^n x^(2n) divided by (2n)!.
What is the radius of convergence for the Maclaurin series of sin x and cos x?
Both have infinite radius of convergence.
What is the Maclaurin series formula for arctan x?
arctan x = sum from n=0 to infinity of (-1)^n x^(2n+1) divided by (2n+1).
What is the Maclaurin series formula for ln(1+x)?
ln(1+x) = sum from n=1 to infinity of (-1)^(n-1) x^n divided by n.
What is the radius of convergence for the Maclaurin series of arctan x and ln(1+x)?
Both have radius of convergence equal to 1.
What is the first step to approximate with a Taylor polynomial?
Choose a center a.
What is the formula for the linear Taylor approximation T₁(x)?
T₁(x) = f(a) + f'(a)(x - a).
How can Taylor approximation error be estimated when a derivative bound is known?
By applying Taylor's Inequality.
What is the second-degree Taylor polynomial for f(x) = x^(1/3) centered at 8?
T₂(x) = 2 + (1/12)(x - 8) - (1/288)(x - 8)².
What error bound was obtained for the second-degree Taylor polynomial of f(x) = x^(1/3) on [7,9]?
|R₂(x)| < 0.0004.
What series is used to approximate (1 + x)^(-1/2) in relativistic kinetic energy?
The binomial series.
What substitution is made in the binomial series for relativistic kinetic energy approximation?
x = -v²/c².
What does relativistic kinetic energy approximate to when v is much less than c?
Approximately (1/2)m₀v².
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