Question

Question 1 - Infinite Sequences.

(a). Determine an infinite sequence that satisfies the following . . .

(i) An infinite sequence that is bounded below, decreasing, and convergent

(ii) An infinite sequence that is bounded above and divergent

(iii) An infinite sequence that is monotonic and converges to 1 as n → ∞

(iv) An infinite sequence that is neither increasing nor decreasing and converges to 0 as n → ∞

(b). Given the recurrence relation an = an−1 + n for n ≥ 2 where a1 = 1, find a explicit formula for an and determine whether the sequence converges or diverges

(c). Find a explicit formula for an given that {an}∞ n=1 generates the infinite sequence 1, − 1 9 , 1 25 , − 1 49 , . . . Does the above infinite sequence converge or divergence?

Answer 1

In the most basic terms, a sequence which is a list of numbers is said to be infinite if it can be mapped to the the infinite sequence of Positive integers i.e (1, 2, 3, 4, ........). So it means that an infinite sequence just goes on continuing forever like the positive integers.

A sequence is said to be bounded, if there exists a real number M such that all elements of the sequence are either greater than M(Bounded Below) or less than M(Bounded Above).

A sequence (a_n) is said to be decreasing if for any i(<n), a_i > a_i+1 .

A sequence (a_n) is said to be increasing if for any i(<n), a_i < a_i+1 .

A sequence is said to converge if the sequence is tending to a fixed value as we go further in the sequence. This fixed value is called the limit value.

A sequence that doesn't converge is said to diverge.

A sequence is said to be monotonic if it is moving in only one direction i.e it is increasing or decreasing. Increasing/Decreasing doesn't mean it has to Increase/Decrease at each step, but if it does Increase/Decrease it has to be the same for a sequence.

(a)
(i) We see that this sequence where a_n = 1/n, tends towards the value 0 as n tends to infinity. We can observe that it is decreasing and is bounded below by 0.

  

(ii) Since any sequence that is not convergent, is divergent a_n = (-1)ⁿ satisfys the given conditions.

  

(iii) Similar to (i) we can have a_n = 1 + 1/n, this sequence is monotonically decreasing and it converges to 1.

  

(iv) A constant sequence of 0 satisfys the given conditions.

(b)

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on adding all these equations we have

We can see that this sequence will diverge because it is tending to infinity as n tends to infinity.