I need with 3! ee Transcribed Image Text: You should explain your reasoning carefully using English sentences where appropriate, not only equations. You
may use the textbook and your notes, and you’re welcome to discuss the problems with one another, with me,
with the TA, but your final answers should be your own and in your own words
In this homework we’ll learn about a new convergence test called Cauchy’s Condensation Test. Suppose n-1 an is
a series where the terms an are > 0 and decreasing, so ant1 < an for all n. The Cauchy condensation test says that n=D1 an converges if and only if 2" azn Converges. n=1 n=1 This test generalizes the idea from Oresme's proof of the divergence of the harmonic series where you group terms together in groups of size 1, 2, 4, 8 etc. 1. By grouping the terms an together into groups of size 1, 2, 4, ..., 2N, argue that 2N+1-1 N Σ an < 2" a a2n. n=1 n=0 2. By taking the limit N o, argue that if E-1 2" azn converges then so does E, an. (Hint: apply the monotone convergence theorem to the sequence of partial sums of the series an.) %=D1 %3D1 This is half of the proof that the Cauchy condensation test works. A similar argument can be used to show that if E 2" azn diverges then so does E=1 an. You don't have to show this. %3D1 %3D1 3. Consider the series EnP log2(n)ª n=2 where p and q are < 0. Use the ordinary comparison test to show that the series converges if p < -1. Here log, means the logarithm with base 2. 4. Use the Cauchy condensation test to show that if p = -1 then this series converges if q <-1 and diverges if q2-1.