Sequences confuse me A sequence is of the form a1, a2, a3, a4, ... where the an

Question

Sequences confuse me

A sequence is of the form a1, a2, a3, a4, ... where the an are real numbers. Technically, a sequence is a function whose domain is the set of natural numbers, and whose range is a subset of the real numbers. Sequences may be defined in various ways: By listing, and appealing (via the three dots) to your intuition. Suppose the sequence is Then the n-th term is an = Explicitly. For example, suppose an = nn. Then a1 = , a2 = , and a3 = , Recursively. For example, the {it Fibonacci Sequence} is defined by a1 = a2 = 1, an+1 = an + an-1, n = 2, 3, 4, .... Thus a3 = , a4 = , and a5 = . A sequence may or may not have a limit. For the following sequences, enter the limit, or enter the letter "D" if the sequence diverges.

Explanation / Answer

1/2, 2/5, 3/10, ... the nth term is = n / (n^2 + 1)

an = n^n, a1 = 1, a2 = 2^2 = 4, a3 = 3^3 = 27, ...

Fibonacci: a3 = a2 + a1 = 1 + 1 = 2, a4 = a3 + a2 = 2 + 1 = 3, a5 = a4 + a3 = 3 + 2 = 5, ...

an = n, Diverges as a(n -> infinity) = infinity (there is no limit)

an = 1/n, Does NOT diverge, a(n -> infinity) = 1/n = 0 (limit exists)

an = {n^2 + 4*n - 5}/{(2*n - 1)*(3*n - 1)}, Does NOT diverge, as limit exists

(Dividing both numerator and denominator by n^2)

a(n -> infinity) = {1 + 4/n - 5/(n^2)} / {(2 - 1/n)*(3 - 1/n)} = 1 / (2*3) = 1/6

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