I need help with 3! Pleaseee You should explain your reasoning carefully using English sentences where appropriate, not only equations. Youmay 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 wordsandIn this homework we'll learn about a new convergence test called Cauchy's Condensation Test. Suppose n-1 an isa series where the terms an are > 0 and decreasing, so ant1 < an for all n. The Cauchy condensation test says thatn=D1an converges if and only if 2" azn Converges.n=1n=1This test generalizes the idea from Oresme's proof of the divergence of the harmonic series where you group termstogether 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 that2N+1-1NΣan < 2" aa2n.n=1n=02. By taking the limit N o, argue that if E-1 2" azn converges then so does E, an. (Hint: apply themonotone convergence theorem to the sequence of partial sums of the series an.)%=D1%3D1This is half of the proof that the Cauchy condensation test works. A similar argument can be used to showthat if E 2" azn diverges then so does E=1 an. You don't have to show this.%3D1%3D13. Consider the seriesEnP log2(n)ªn=2where p and q are < 0. Use the ordinary comparison test to show that the series converges if p < -1. Herelog, 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 ifq2-1.

Name: I need help with 3! Pleaseee You should explain your reasoning careful
Uploaded: 2024-03-20T07:06:27.000Z
Channel: Bartleby
Description: I need help with 3! Pleaseee You should explain your reasoning carefully using English sentences where appropriate, not only equations. Youmay use the textbook and your notes, and you're welcome to discuss the problems with one another, with me,with