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1. Discuss convergence or divergence for each of the following series: 00 2 (2j)! (b) (3j)! j=1 8 (-1)³ (c) 80 (d) 3j² - 5j + 6 j =1 3 2j-1 - (f) 3j³-2 - j=1 3. If bj > 0 for every j and if [jo bj converges then prove that Ejo =1 (bj) 2 converges. Prove that the assertion is false if the posi- tivity hypothesis is omitted. How about third powers? 4. If b, > 0 for every j and if jo b, converges then prove that = 1 diverges. j and if [jo b, converges then prove that C, = bj converges. 1+bj 12. Follow these steps to give another proof of the Alternating Se- ries Test: a) Prove that the odd partial sums form an increasing sequence; b) Prove that the even partial sums form a decreasing sequence; c) Prove that every even partial sum majorizes all sub- sequent odd partial sums; d) Use a pinching principle.

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