Chapter 7: Q. 19 (page 615)
Find two divergent geometric series and with all positive terms such that converges.
Short Answer
Ans:
part (a). The divergent geometric series
part (b). The divergent geometric series
part (c). The series is converge
Chapter 7: Q. 19 (page 615)
Find two divergent geometric series and with all positive terms such that converges.
Ans:
part (a). The divergent geometric series
part (b). The divergent geometric series
part (c). The series is converge
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Get started for freeGiven a series , in general the divergence test is inconclusive when . For a geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.
Let be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to . (Hint: Argue that if you add up some finite number of the terms of , the sum will be greater than . Then argue that, by adding in some other finite number of the terms of
, you can get the sum to be less than . By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to .)
Determine whether the series converges or diverges. Give the sum of the convergent series.
Given thatand, find the value of.
Determine whether the series converges or diverges. Give the sum of the convergent series.
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