1. Discuss convergence or divergence for each of the following series:
3j² - 5j + 6
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.
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.
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.