The following pictures contains three exercises about series that I resolved and would like to know if they're correct. I tried resolving them with Symbolab, but the app solves the exercises with other criterions and would like to know if they way I worked them are correct.
I've attached two pictures, the first one is the exercise labeled 22, which I solved with the Alternating Series criterion:
Exercise 22 - Alternating Series Criterion
And the other one contains two exercises - 68 and 24 which I solved with the Quotient (Ratio?) Criterion and the Alternating Series Criterion, respectively:
Exercises 68, 24 - Quotient and Alternating Series criterions, respectively
I appreciate any input and feedback but also apologize for the Spanish. However, I'm glad that math is an universal language.
Oh, btw... My conclusion was that all three series converged.
If missed important information or did something wrong please let me know.
Thank you for your help!
ex. 22, the alternating series test told you the alternate series converges, but you still have to check if this an "absolute convergence" or a "conditional convergence". (not sure they are called that, I'm brazilian and not used to writing math in english, hehe).
ex 68, looks right! sorry about first answer! edited this.
ex 24, same as 22, you apply the alternate series test just right, but gotta go until the end and test the absolute series to check if it's an absolute or a conditional convergence.
(Obs.: To test the absolute series, you will ignore the alternating factor on those series [(-1) to the n, n+1, etc] and treat them as if they were regular positive series, using one of the usual tests (ratio, comparison, root, etc.)