How do you find the boundedness of a sequence?
A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K’, greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K’.
How do you know if a sequence is convergent?
If limn→∞an lim n → ∞ exists and is finite we say that the sequence is convergent. If limn→∞an lim n → ∞ doesn’t exist or is infinite we say the sequence diverges.
What is convergence of sequence?
A sequence converges when it keeps getting closer and closer to a certain value. Example: 1/n. The terms of 1/n are: 1, 1/2, 1/3, 1/4, 1/5 and so on, And that sequence converges to 0, because the terms get closer and closer to 0. (Also called “Convergent Sequence”)
How do you know if its bounded above or below?
Being bounded from above means that there is a horizontal line such that the graph of the function lies below this line. Bounded from below means that the graph lies above some horizontal line. Being bounded means that one can enclose the whole graph between two horizontal lines.
How do you know if a sequence is bounded or unbounded?
A sequence an is a bounded sequence if it is bounded above and bounded below. If a sequence is not bounded, it is an unbounded sequence. For example, the sequence 1/n is bounded above because 1/n≤1 for all positive integers n. It is also bounded below because 1/n≥0 for all positive integers n.
How do you determine if a series is convergent or divergent?
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.
How do you know if a function is convergent or divergent?
convergeIf a series has a limit, and the limit exists, the series converges. divergentIf a series does not have a limit, or the limit is infinity, then the series is divergent.
What is convergent sequence with example?
For an example of a convergent sequence, let us examine an=(1+1n)n, the well known sequence that converges to e, Euler’s number. an=3n4+34n3+142n2+15n+8 is a divergent sequence. This is clear because the expression is “top-heavy” because the degree of the numerator is greater than that of the denominator.
What is a monotone sequence?
We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value. Of course, sequences can be both bounded above and below.