Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent \n+1 to 0, then the series >(-1)n+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the n=1 result still true? Consider the series defined by 8. 1 1 + 16 1 1 1 - + 4 1 -- + 3 1 1 E(-1)"+1bn; - - .. 8. 4 n 2n n=1 where the sequence b, is defined by 1 1 1 1 1 1 1 1 1 1 2'4'3'8'4' 16'5' 32 2n n (a) Does this sequence {b„} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and which does it fail? 1 Σ E(;-). Suppose that it did (b) Show that this series diverges. (Hint: This series can also be written as 2n n=1 converge. If you add a certain geometric series to it, you're adding two convergent series together, so you should get another convergent series-but do you?) (c) Is the Alternating Series Test wrong? Explain why not.

Determine whether or not the following series converge

1. In this problem, you'll look at the series > 1 )J: One can visualize this series as area, by drawing a k5/4 * k=10 box of area 103/4, a box of area 134 and so on. 115/4 (a) There are two ways to draw the series (as boxes) in question. Draw each of them together with the graph y= 34 on two different graphs. x5/4 (b) Use the picture to write an inequality comparing >) 1 to an improper integral. Check the k5/4 k=10 bounds of your improper integral carefully, and be sure you are comparing on the same domain. (c) Is the improper integral you wrote in (b) convergent or divergent? Explain. (d) Using the inequality you wrote in (b) and your answer to (c), what (if anything) can you conclude 1 :? about the convergence of 2 k5/4 k=10 (e) Repeat (b)-(d) with your second picture.

Write out the first 5 terms of the power series (3)" п! n=0