Does the sequence sin n have a convergent subsequence Why?

Does the sequence sin n have a convergent subsequence Why?

Theorem Bolzano-Weierstrass Every bounded sequence has a convergent subsequence. Example The weird, oscillating sequence (sin n) is far from being convergent. But, since −1 ≤ sin n ≤ 1, we are guaranteed that it has a convergent subse- quence.

Why does a sequence converge?

A sequence is “converging” if its terms approach a specific value as we progress through them to infinity.

Is the sequence sin n )/ n convergent?

yes. |sin n| ≤ 1 for all natural number n. so the given series is absolute convergent .

Is the sequence sin n )} bounded does it converge?

Thus the sequence (sin(n))n does not converge.

Is sin n/a bounded sequence?

we know that this is bounded but isn’t convergence. We know {a_n} have a subsequence which is convergence. …

Does the sequence n converge?

If we say that a sequence converges, it means that the limit of the sequence exists as n → ∞ n\to\infty n→∞. If the limit of the sequence as n → ∞ n\to\infty n→∞ does not exist, we say that the sequence diverges. If the limit exists then the sequence converges, and the answer we found is the value of the limit.

What does it mean for a sequence to converge or diverge?

A sequence is said to be convergent if it approaches some limit (D’Angelo and West 2000, p. 259). Formally, a sequence converges to the limit. if, for any , there exists an such that for . If does not converge, it is said to diverge.

Does sin n n converge absolutely?

yes. |sin n| ≤ 1 for all natural number n. so the given series is absolute convergent . so the series is convergent.

What does sin n n converge?

0
sin n/√n = 0 and the sequence converges to 0. every term is 1, and likewise if r = 0 the sequence converges to 0.

Does the sum of sin n converge?

yes. |sin n| ≤ 1 for all natural number n.

Does the sequence (sin(n))n converge?

Thus the sequence (sin(n))n does not converge. Remark. A variation on this proof shows that the set {sinn: n ∈ N} is dense in the interval [ − 1, 1]. Not the answer you’re looking for?

How do you know if a sequence converges or diverges?

If we say that a sequence converges, it means that the limit of the sequence exists as n → ∞ ntoinfty n → ∞. If the limit of the sequence as n → ∞ ntoinfty n → ∞ does not exist, we say that the sequence diverges. A sequence always either converges or diverges, there is no other option.

How do you use the squeeze theorem to determine convergence?

Sometimes it’s convenient to use the squeeze theorem to determine convergence because it’ll show whether or not the sequence has a limit, and therefore whether or not it converges. Then we’ll take the limit of our sequence to get the real value of the limit.

What is the value of N → ∞ in sin(n + 1)?

Letting n → ∞ in sin(n + 1) = sinncos1 + cosnsin1 would then imply cos1 = 1, which is clearly wrong. This needs a deeper result. Let the fractional part of x be denoted by {x}. By Kronecker’s Theorem, the set {{ n 2π}: n ∈ N} is a dense subset of [0, 1], (because π is irrational.)