Chapter 7: Q. 1TF (page 626)
Q.
A series of monomials: Find all values of \(x\) for which the series \( \sum_{k=1}^{∞} (4x)^k\) converges.
Short Answer
The value of \(x\) lies in the interval \(\left (\frac{-1}{4},\frac{1}{4} \right )\)
Chapter 7: Q. 1TF (page 626)
Q.
A series of monomials: Find all values of \(x\) for which the series \( \sum_{k=1}^{∞} (4x)^k\) converges.
The value of \(x\) lies in the interval \(\left (\frac{-1}{4},\frac{1}{4} \right )\)
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Get started for freeExpress each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.
Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.
35.
Find the values of x for which the series converges.
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.
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