Answer:
- d, series and sequence diverge
- d, geometric/divergent
- c, e (geometric, |r|<1)
Step-by-step explanation:
<h3>1.</h3>
The sequence terms have a common difference of -5/8. That make it a non-trivial arithmetic sequence, so it diverges.
The series is the sum of terms of the sequence. Any non-trivial arithmetic series diverges.
(A "trivial" arithmetic series has a first term of 0 and a common difference of 0. It is the only kind of <em>arithmetic</em> series that doesn't diverge.)
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<h3>2.</h3>
The terms of the series have a common ratio of -2. That makes it a geometric series. The ratio magnitude is greater than 1, so the series diverges.
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<h3>3.</h3>
A sequence will converge only if the terms have a common difference of 0 or a common ratio with a magnitude less than 1. Of the offered choices, only C and E will converge:
c. geometric, r = 3/5
e. geometric, r = -1/6
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<em>Additional comment</em>
The convergence criteria stated for problem 3 applies only to arithmetic and geometric sequences. There are many other kinds of sequences that converge, but these are the kinds being considered in this problem set.
