I don't even know where to start or go with this first problem. A) Assume that 'a sub n' E C and consider rearrangements of the convergent series the 'sum of 'a sub n' from n=1 to infinity'. Show that each of the following situations is possible and that this list includes all possibilities. 1) The sum of a sub n from n=1 to infinity converages absolutely and hence all rearrangements converge to the same value. 2) The set of possible values of convergent rearrangements is all of C. 3) The set of possible values of convergent rearrangements is an arbitrary line in C. B) Show that the sum of [ (-1)^(n+1) / n ] from n=1 to infinity converges conditionally. The value of the sum is log(2), approcimately .69. Show that by grouping the terms by taking two positive terms, then one negative term, then two positive terms, then one negative term, and so on, the series adds up to a number larger than 1. The value of the sum therefore depends on the order in which the terms are summed. This is all that I could come up with: The abs value of the series diverges because it's the harmonic series where p=1. The series itself converges by the alternating series test because the nth term converges to zero and the series is decreasing. Therefore the given series converges conditionally. Now for the rearranging: 1 + 1/3 -1/2 + 1/5 + 1/7 -1/4 and so forth. I then just listed the partial sums. s1= 1 + 1/3 -1/2= 5/6 s2= s1 + 1/5 + 1/7 -1/4= 0.926 s3= s2 + 1/13 + 1/15 - 1/8= 0.98013 and so on... I know that this isn't right. I don't know how I'm supposed to show that the sum of the rearranged series is greater than 1. C) Express the rearranged series *found above in B* in the form 'the sum of b sub n from n=1 to infinity', and find a simple expression for b sub n. Try to find the infinite sum. After a lot of erasing I came up with: 1 + 1/3 -1/2 + 1/5 + 1/7 -1/4 and so forth can be expressed as the sum of 1/ (4k-3) + 1/(4k-1) - 1/(2k) from k=1 to infinity. I don't know if this is correct for 'b sub n' form or how to find the infinite sum of this rearranged series. D) Assume that 'a sub n' E R(real) and that the sum of 'a sub n' from n=1 to infinity converged conditionally. Fix an arbitrary L(limit) E R(real). Show that we can rearrange the terms to make the sum converge to L. I don't know where to start or where to go with this one, too. Do I treat 'a sub n' as an alternating series that converges conditionally? I know that you aren't here to solve it for me, but any help you can give me would be great. Thank you very much. Please, let me know if I need to clarify anything.