Here are a some sequences of numbers:a) [math: 1,−1,1/2,−1/2,1/3,−1/3,…]1, -1, 1/2, -1/2, 1/3, -1/3, \ldotsb) [math: 1,−2,3,−4,5,−6,…]1, -2, 3, -4, 5, -6, …c) [math: 0,0,0,0,0,…]0, 0, 0, 0, 0, …d) [math: 1,2,3,4,5,6,…]1, 2, 3, 4, 5, 6, …e) [math: 0+1,1+1/2,0+1/3,1+1/4,0+1/5,1+1/6,….]0+1, 1+1/2, 0+1/3, 1+1/4, 0+1/5, 1+1/6, ….How many of these sequences are divergent?

Question

BlackTom AI · Accepted Answer

Here is a step-by-step analysis of each sequence to determine whether it is divergent or convergent.

Option a) [1, −1, 1/2, −1/2, 1/3, −1/3, ...]
- This sequence alternates in sign and the absolute value of the terms is 1, 1, 1/2, 1/2, 1/3, 1/3, ... which decreases to 0 as the index increases. Since the terms approach 0 and the signs alternate in a controlled, decreasing magnitude, the sequence has a finite limit (0). Therefore, it is not divergent.

Option b) [1, −2, 3, −4, 5, −6, ...]
- The absolute values are 1, 2, 3, 4, 5, 6, ... which grow without bound. The sequence does not approach any finite number and does not settle into a pattern that would yield a limit. Hence, this sequence is divergent.

Option c) [0, 0, 0, 0, 0, ...]
- Every term is 0, so the sequence obviously converges to 0. There is a definite finite limit, so it is not divergent.

Option d) [1, 2, 3, 4, 5, 6, ...]
- This sequence increases without bound and does not approach any finite limit. It diverges to infinity in the conventional sense, so it is divergent.

Option e) [0+1, 1+1/2, 0+1/3, 1+1/4, 0+1/5, 1+1/6, ...]
- Writing more plainly, the sequence terms are 1, 1.5, 1/3, 1.25, 1/5, 1.166..., alternating between values of the form 0+1/n and 1+1/n. The odd-indexed terms are 1/n with n odd, which tend to 0 as n grows, while the even-indexed terms are 1 + 1/n with n even, which tend to 1. Because the sequence has two distinct subsequential limits (0 and 1), it does not converge to a single finite value. Therefore, it is divergent.

In summary, after assessing each option, the divergent sequences are those whose terms do not settle to a single finite limit: options b, d, and e are divergent, while a and c are convergent.

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