Let
{
𝑎
𝑛
}
𝑛
=
0
∞
be a sequence.

Which statements are equivalent to "
{
𝑎
𝑛
}
𝑛
=
0
∞
is divergent to
∞
" ?

Select all the correct answers.

Question

Let 
{
𝑎
𝑛
}
𝑛
=
0
∞
 be a sequence.

Which statements are equivalent to "
{
𝑎
𝑛
}
𝑛
=
0
∞
 is divergent to 
∞
 " ?

Select all the correct answers.

BlackTom AI · Accepted Answer

To understand what it means for a sequence {a_n} to diverge to +∞, we need a definition that ties all large n to arbitrarily large positive values. 
Option 1: ∀ M ∈ N, ∃ n0 ∈ N, ∀ n ∈ N, n > n0 ⇒ a_n > M. This matches the standard formal definition: for every threshold M (taken here from natural numbers, which is fine since natural numbers are unbounded), there exists a point n0 after which all terms exceed M. In other words, eventually the sequence stays above every fixed level M. This is a correct formulation for divergence to +∞. 
Option 2: ∀ M > 0, ∃ n0 > 0, ∀ n ∈ N, n > n0 ⇒ |a_n| > M. Here we demand that the magnitude of a_n eventually exceeds every positive M. While this ensures the sequence becomes large in magnitude, it does not guarantee a_n itself goes to +∞; the terms could alternate sign (e.g., a_n could go to -∞ or oscillate with large magnitude). Therefore, this statement is not equivalent to divergence to +∞. 
Option 3: ∀ M > 0, ∃ n ∈ N, a_n > M. This says that for every threshold M, there exists at least one index n (not necessarily after some fixed point) with a_n > M. However, it does not require that all sufficiently large n satisfy a_n > M. A sequence could reach arbitrarily large values at some scattered indices but fail to stay above M eventually. Hence this is weaker than the definition and not equivalent to divergence to +∞. 
Option 4: ∀ M > 0, ∃ n0 ∈ N, ∃ n0 > 0, ∀ n ∈ N, n > n0 ⇒ a_n > M. Translating the intent, this says that for every M there is some n0 > 0 so that all later terms exceed M. The extra requirement n0 > 0 does not weaken the condition; it still asserts that beyond some positive index, every subsequent term is above M. This aligns with the standard definition and is therefore equivalent to divergence to +∞. 
In summary, the statements that correctly capture divergence to +∞ are those that force all sufficiently large terms to exceed every fixed threshold M, not just occasionally or in magnitude. The first and the fourth options express that precise idea, while the second uses absolute value and the third allows only occasional exceedances rather than eventual dominance.

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 50 (11.1 and 11.2)

Let { 𝑎 𝑛 } 𝑛 = 0 ∞ be a sequence. Which statements are equivalent to " { 𝑎 𝑛 } 𝑛 = 0 ∞ is divergent to ∞ " ? Select all the correct answers.

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