Question

Consider the following series:

$\sum_{k=1}^{\infty } \frac{5(-1)^{k}}{3^{k}}$

We would like to analyze it to figure out convergence/divergence.

1. What test did you use, or what type of series is this?
2. What was your reasoning for using that test or noticing that type of series?
3. Show some work (labels, write out relevant limits, etc.)
4. What are your conclusions? Does the series diverge? Does it converge? If it is relevant, does it converge conditionally or absolutely? If it is relevant, what does it converge to?

Answer 1

~ Solution (a Consider the senes e souk 3K This is an alterna emating series of the form I (kak Ž Gokak where K-1 $ al= 3K series is Lei britz for Alternating Then the conver gence test which says uk given by 1 akso (1 akti if Lak ak converges (111) lim R=1 9k=0. Kw kt < ak- and for KEN , 3 > 3 9 KH 3 3K 3K 1.5 and 5 < - akti Lak , So sequence is decreasing. 3* 3kt K 5 Now him ax= converges lim KYN 3K ka 3K ko {( ak = SGK by Leibnitzlest. let us check for absolute convergence, Then for Ak= s Now, let 5 돐 E ak = K=1 E / 3k Apply Ratic test for 3K Convergence or diver gence, let 3K 5 akt 3. ak 3K+1.S ie sus ant Tak converges Ka 3K K-1 Ka ak overges absolutely o prenel

b Now let us compute its sum ; So let I brak Z 5 ka Euk 3K which is also K여 sa Guk Ra 3k metric series, a geometrics It t- u Ž Ellak = sl -| 3 Xk 3k 32 33 K=1 [:a. a atartart. 1-(-3) Eco Wars - cuxax = 5() - 413 o Gokak e , sum of Series 1 ( tako se pred