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- A sequence is said to be bounded above if there exists a real number M such that every term in the sequence is less than or equal to M. The smallest such M is called the least upper bound or supremum.
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Does a bounded series converge?
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No, a bounded series does not necessarily converge. Consider the series ∑(− 1)n (heavily related to Henning's example). It will forever oscillate between 0 and 1 (or -1 and 0, depending on the indices). But if the partial sums are bounded and monotonic, then it does converge.
- How to prove a sequence is bounded above or below
How to prove the sequence $X_n=(1+X_{n-1})/2$ is...
- How to prove a sequence is bounded above or below
Nov 5, 2017 · How to prove the sequence $X_n=(1+X_{n-1})/2$ is bounded-above or -below given $X_0$ 1 Prove that a sequence is bounded if and only 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} {1 n} is bounded above because 1 n ≤1 1 n ≤ 1 for all positive integers n n. It is also bounded below because 1 n ≥0 1 n ≥ 0 for all positive integers n. Therefore, {1 n} {1 n} is a bounded sequence.
Dec 21, 2020 · A sequence \(\displaystyle {a_n}\) is bounded above if there exists a real number \(\displaystyle M\) such that \(\displaystyle a_n≤M\) for all positive integers \(\displaystyle n\).
Jul 11, 2023 · We will also determine a sequence is bounded below, bounded above and/or bounded. Series – The Basics – In this section we will formally define an infinite series. We will also give many of the basic facts, properties and ways we can use to manipulate a series.
A sequence {a n} is said to be bounded above if there exists an M such that a n < M for all n in ℕ; it is bounded below if there exists an m such that m < a n for all n in ℕ.
If you just want the bounded property, you can simply find crude upper and lower bounds. E.g. if (−1)n + 1/n (− 1) n + 1 / n is bounded, then so is ((−1)n + 1/n)2 ((− 1) n + 1 / n) 2 (so we can essentially disregard the power of 2).
