An Improper Integral and Infinite Series: Sketch the function for x ≥ 1 together with the graph of the terms of the series Argue that for every term of the sequence of partial sums for this series,. What does this result tell you about the convergence of the series?
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 .)
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