What is an infinite sequence in mathematics?
An ordered list of numbers indexed by specified positive integers.
What does the notation \(a_n \to L\) as \(n \to \infty\) signify?
The terms get arbitrarily close to \(L\) for large \(n\).
What does a finite limit imply about a sequence?
That the sequence converges.
When does the sequence \(\{r^n\}\) converge?
Precisely when \(-1 < r \leq 1\).
What is the limit of \(\{r^n\}\) if \(-1 < r < 1\)?
The limit is zero.
What is the limit of \(\{r^n\}\) when \(r = 1\)?
The limit is one.
What does the Monotonic Sequence Theorem state about bounded monotonic sequences?
They always converge.
What types of bounded monotonic sequences does the Monotonic Sequence Theorem specify?
Increasing sequences bounded above or decreasing sequences bounded below.
What defines an infinite series in terms of partial sums?
An infinite series is defined by its partial sums $s_n=\sum_{i=1}^n a_i$.
When does an infinite series converge?
It converges when its partial sums approach a finite real number.
What is the sum formula for a geometric series with $|r|<1$?
The sum is $\frac{a}{1-r}$.
What condition on $r$ ensures convergence of a geometric series?
The series converges if $|r|<1$.
What does the Test for Divergence state about $\lim_{n\to\infty} a_n$?
If the limit does not exist or is not zero, the series diverges.
What happens to a series if $\lim_{n\to\infty} a_n$ is not zero?
The series diverges by the Test for Divergence.
Does the harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ converge?
No, the harmonic series diverges.
Do the terms of the harmonic series approach zero?
Yes, its terms approach zero.
When does the Integral Test apply to a series?
When a_n = f(n) with f continuous, positive, and decreasing on the tail.
What does the Integral Test conclude about series and integrals?
The series and integral either both converge or both diverge.
When does the p-series ∑ 1/n^p converge?
It converges when p > 1.
When does the p-series ∑ 1/n^p diverge?
It diverges when p ≤ 1.
What inequality bounds the remainder R_n for a convergent positive decreasing series?
∫_{n+1}^∞ f(x) dx ≤ R_n ≤ ∫_n^∞ f(x) dx.
How many terms are needed for ∑ 1/n^3 to have error below 0.0005?
At least 32 terms are needed.
What remainder bound is used for ∑ 1/n^3 to estimate error?
R_n ≤ 1/(2n^2).
When does Direct Comparison prove convergence for positive terms?
When 0 ≤ a_n ≤ b_n and ∑b_n converges.
When does Direct Comparison prove divergence for positive terms?
When a_n ≥ b_n ≥ 0 and ∑b_n diverges.
What condition must hold for Limit Comparison to apply to positive-term series?
The limit of a_n/b_n as n→∞ equals a finite positive constant c.
What conclusion does Limit Comparison give about two series with positive terms?
They have the same convergence behavior.
How do you compare rational or algebraic terms to a known p-series?
By comparing dominant powers of n using Direct or Limit Comparison.
What inequality relates remainders of series when 0 ≤ a_k ≤ b_k for large k?
The remainder of the a_k series is no larger than that of the b_k series.
When does the Alternating Series Test prove convergence?
When the terms decrease and approach zero.
What inequality bounds the remainder in an alternating series satisfying the test?
The remainder's absolute value is at most the next term's magnitude.
What defines absolute convergence of a series?
The series of absolute values converges.
What defines conditional convergence of a series?
The series converges but its absolute value series diverges.
What does absolute convergence imply about ordinary convergence?
Absolute convergence implies ordinary convergence.
What is true about rearrangements of absolutely convergent series?
They all have the same sum.
What is the first step in selecting a convergence test?
Inspect the term limit.
Which series forms should you check after the term limit?
P-series or geometric form.
What should you inspect after checking for p-series or geometric form?
Signs and algebraic structure.
Which test is suggested for factorials, products, or constant-to-the-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 useful for many rational or p-series terms?
Because a_{n+1}/a_n tends to 1.
What is the geometric series identity for |x|<1?
1/(1-x) equals the sum from n=0 to infinity of x^n.
What series results from substituting -x^2 into the geometric series?
1/(1+x^2) equals the sum from n=0 to infinity of (-1)^n x^{2n}.
What operation can be done term by term inside a power series' radius of convergence?
A power series can be differentiated or integrated term by term inside its radius of convergence.
What happens to the radius of convergence after term-by-term differentiation or integration?
The resulting series has the same radius of convergence.
What is the domain of the Bessel-function power series example?
Its domain is all real numbers because it converges for every real x.
Can the Bessel-function power series be differentiated term by term?
Yes, it can be differentiated term by term.
What is the formula for the coefficient in a Taylor series centered at a?
It is c_n = f^(n)(a) divided by n!
How is the Taylor series centered at a expressed as a sum?
As the sum from n=0 to infinity of (f^(n)(a)/n!) times (x - a)^n
What defines a Maclaurin series in relation to a Taylor series?
It is a Taylor series centered at a = 0
How is a Maclaurin series written as a sum?
As the sum from n=0 to infinity of (f^(n)(0)/n!) times x^n
What is the nth-degree Taylor polynomial T_n(x)?
It is the finite sum from i=0 to n of (f^(i)(a)/i!) times (x - a)^i
Does the nth-degree Taylor polynomial always equal the function?
No, it need not equal the function
When does a function equal its Taylor series on an interval?
Only when the remainder R_n(x) approaches zero there
What is the remainder R_n(x) in Taylor series approximation?
It is f(x) minus the nth-degree Taylor polynomial T_n(x)
What is the Maclaurin series formula for the exponential function?
It is 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 radius of convergence do sin x and cos x have in their standard series?
They have an infinite radius of convergence.
What radius of convergence do arctan x and ln(1+x) have in their standard series?
They have a radius of convergence equal to 1.
What is the binomial series formula for (1+x)^k?
It is (1+x)^k = sum from n=0 to infinity of binomial(k,n) times x^n.
How is the binomial coefficient binomial(k,n) defined for the binomial series?
It is k(k-1)...(k-n+1) divided by n!.
What is the domain condition for the binomial series (1+x)^k to converge?
It converges for |x| less than 1.
Name methods to generate new series from known series.
Substitution, multiplication, term-by-term differentiation or integration, formal multiplication, and power-series long division.
What is the formula for the linear Taylor approximation T₁(x)?
T₁(x) = f(a) + f'(a)(x - a).
What does Taylor’s Inequality bound in terms of the remainder Rₙ(x)?
|Rₙ(x)| ≤ (M|x - a|^{n+1}) / (n+1)! when |f^{(n+1)}(x)| ≤ M.
What is the second-degree Taylor polynomial for f(x) = x^{1/3} at a = 8?
T₂(x) = 2 + (1/12)(x - 8) - (1/288)(x - 8)^2.
What error bound is given for the cube-root approximation on 7 ≤ x ≤ 9 using M = 0.0021?
|R₂(x)| < 0.0004.
What classical approximation results from expanding relativistic kinetic energy for v ≪ c?
K ≈ (1/2) m₀ v².
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1. What distinguishes a sequence from a series?
2. What does the statement $a_n\to L$ as $n\to\infty$ mean?
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