An ordered list of numbers indexed by integers
Explication
An infinite sequence is an ordered list of numbers, commonly written with nth term a_n, and it can be viewed as a function whose inputs are integers. A series, in contrast, is a sum of terms.
An ordered list of numbers where each term is defined by a specific rule.
Explication
A sequence is an infinite ordered list of numbers, typically represented as a_n, where each term is associated with an integer index. The other options do not accurately describe the concept of a sequence.
For -1 < r ≤ 1
Explication
The sequence r^n converges precisely when -1 < r ≤ 1. Its limit is 0 when -1 < r < 1 and 1 when r = 1.
An ordered list of numbers where the nth term is written as a_n
Explication
A sequence is an ordered list of numbers, with each term typically denoted as a_n, representing a function from the integers to the real numbers.
It is bounded and monotonic
Explication
Every bounded monotonic sequence converges. Boundedness alone does not guarantee convergence, and monotonicity alone does not guarantee it either.
To determine whether the sequence converges or diverges
Explication
Analyzing monotone and bounded sequences helps determine their convergence or divergence, as every bounded monotonic sequence converges.
1
Explication
Since n/(n+1) = 1/(1+1/n) and 1/n approaches 0, the denominator approaches 1, so the sequence converges to 1.
In the early 19th century with the formalization of series convergence criteria
Explication
The integral and P-series tests were developed as part of the formalization of convergence criteria in the 19th century, particularly with the work of mathematicians like Cauchy and others who contributed to rigorous analysis.
Comparison tests determine convergence by comparing series terms to known series, while error bounds estimate the difference between the partial sum and the actual sum.
Explication
Comparison tests analyze the convergence of a series by comparing it to a known convergent or divergent series, whereas error bounds estimate how close a partial sum is to the actual sum, often using integral estimates.
It helps determine the series' convergence or divergence.
Explication
Choosing the right convergence test allows us to accurately determine whether an infinite series converges or diverges, which is essential for proper analysis. It does not necessarily guarantee convergence or give the sum, but it guides the decision.
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What is an infinite sequence in mathematics?
An infinite sequence is an ordered list of numbers indexed by integers.
Sequence limit
Sequence terms approach L as n→∞.
When does a sequence a_n converge to a limit L?
When its terms become arbitrarily close to L as n becomes large.
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