Examples of sequence in the following topics:
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- A sequence is an ordered list of objects (or events).
- Also, the sequence (1,1,2,3,5,8), which contains the number 1 at two different positions, is a valid sequence.
- Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2,4,6,⋯).
- Finite sequences are sometimes known as strings or words, and infinite sequences as streams.
- The empty sequence () is included in most notions of sequence, but may be excluded depending on the context.
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- A series is the sum of the terms of a sequence.
- Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
- The sequence of partial sums Sk associated to a series ∑n=0∞an is defined for each k as the sum of the sequence an from a0 to ak:
- Infinite sequences and series can either converge or diverge.
- An infinite sequence of real numbers shown in blue dots.
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- A series is the sum of the terms of a sequence.
- A series is, informally speaking, the sum of the terms of a sequence.
- Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
- Given an infinite sequence of numbers {an}, a series is informally the result of adding all those terms together: a1+a2+a3+⋯ .
- The sequence of partial sums {Sk} associated to a series ∑n=0∞an is defined for each k as the sum of the sequence {an} from a0 to ak:
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- Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
- Proof: Suppose the sequence an converges to 0 and is monotone decreasing.
- (The sequence {Sm} is said to form a Cauchy sequence, meaning that elements of the sequence become arbitrarily close to each other as the sequence progresses.)
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- For a sequence {an}, where an is a non-negative real number for every n, the sum ∑n=0∞an can either converge or diverge to ∞.
- For a sequence {an}, where an is a non-negative real number for every n, the sequence of partial sums
- Therefore, it follows that a series ∑n=0∞an with non-negative terms converges if and only if the sequence Sk of partial sums is bounded.
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- In this example, the ratio of adjacent terms in the blue sequence converges to L=21.
- Then the blue sequence is dominated by the red sequence for all n≥2.
- The red sequence converges, so the blue sequence does as well.
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- A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
- Identify a transcendental function as one that cannot be expressed as the finite sequence of an algebraic operation
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- Since the sum of the sequence n1 (i.e., ∑n1) diverges, the limit convergence test tells that the original series (with 2n2n+1) also diverges.
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- The above examples involving the harmonic series raise the question of whether there are monotone sequences such that f(n) decreases to 0 faster than n1but slower than n1+ε1 in the sense that:
- Once such a sequence is found, a similar question can be asked of f(n) taking the role of n1 oand so on.
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- However, f itself is not continuous as can be seen by considering the sequence f(n1,n1) (for natural n), which should converge to f(0,0)=0 if f is continuous.