Thursday, March 26, 2020

Advanced Math Solutions - Series Convergence Calculator, Series Ratio Test

In our Series blogs, we’ve gone over four types of series, Geometric, p, Alternating, and Telescoping, and their convergence tests. Now, we will focus on convergence tests for any type of infinite series, as long as they meet the tests’ criteria. We will start off with the Series Ratio Test.

What does the Series Ratio Test do?

The goal of the Series Ratio Test is to determine if the series converges or diverges by evaluating the ratio of the general term of the series to its following term. The test determines if the ratio absolutely converges. A series absolutely convergences if the sum of the absolute value of the terms is finite. If there is absolute convergence, then there is convergence. This will make more sense, once you see the test and try out a few examples.

Some caveats: The test will not determine what the series will converge to. The test may also result in inconclusive results.

What is the Series Ratio Test?

Given a series, ∑a_n , determine L such that

                                             lim_{n→∞} \mid\frac{a_{n+1}}{a_n}\mid=L,where a_n≠0
If L<1, then ∑a_n  converges.
If L>1, then ∑a_n  diverges.
If L=1, then the test is inconclusive.

The test may seem pretty straight forward and simple, but determining what type of series to use this test on is not.

What types of series should the test be used on?

While there is no straight forward answer to this question, this test is typically helpful when determining convergence for series with exponential functions or factorials.

Now that we know what the series ratio test is, let’s see some examples of how it is used.

First example (click here):

                                                             ∑_{n=0}^∞\frac{2^n}{n!}

1. Determine the limit of lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid

                 lim_{n→∞} \mid\frac{\frac{2^{n+1}}{(n+1)!}}{\frac{2^n}{n!}}\mid=lim_{n→∞} \mid\frac{2^{n+1}∙n!}{(n+1)!∙2^n }\mid
                                                   =lim_{n→∞} \mid\frac{2∙n!}{(n+1)!}\mid

                                                             (Note: 2^{n+1} = 2^n ∙ 2^1 )

                                                         =lim_{n→∞} \mid\frac{2}{(n+1) }\mid

                                                (Note: \frac{n!}{(n+m)!}=\frac{1}{(n+1)∙(n+2)  ∙ ∙ ∙(n+m) })

                                                                         =0=L

For detailed procedures on how to determine the limit, please see the step by step procedures in the link.

2. Given L, determine convergence

                                                 Since L = 0 and is < 1, the series converges.

Next example (click here):

                                                                  ∑_{n=1}^∞\frac{6^n}{n}

1. Determine the limit of lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid

                  lim_{n→∞} \mid\frac{\frac{6^{n+1}}{n+1}}{\frac{6^n}{n}}\mid=lim_{n→∞} \mid\frac{6^{n+1}∙n}{(n+1)∙6^n }\mid

                                                        =lim_{n→∞} \mid\frac{6n}{(n+1) }\mid

                                                       =6∙lim_{n→∞} \mid\frac{1}{(1+\frac{1}{n}) }\mid

                                                            (Note: \frac{n}{n+1 }=  \frac{1}{1+\frac{1}{n}})

                                                                         =6∙1=6=L

2. Given L, determine convergence

                                                  Since L = 6 and is > 1, the series diverges.


Next example (click here):

                                                                 ∑_{n=0}^∞\frac{2^2n}{3^2n}

1. Determine the limit of lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid

 lim_{n→∞} \mid\frac{\frac{2^{2(n+1)}}{3^{2(n+1)}} }{\frac{2^2n}{3^2n} }\mid=lim_{n→∞} \mid\frac{(\frac{2}{3})^{2(n+1)}}{(\frac{2}{3})^2n} \mid

                                                       =lim_{n→∞} \mid(\frac{2}{3})^{2(n+1)-2n} \mid

                                                               =lim_{n→∞} \mid(\frac{2}{3})^2 \mid

                                                                               =\frac{4}{9}=L

2. Given L, determine convergence

                                                    Since L = \frac{4}{9} and is < 1, the series converges.

Last example:
                                                                      ∑_{n=1}^∞\frac{1}{n}


1. Determine the limit of lim_{n→∞} \mid\frac{a_{n+1}}{a_n} \mid

                                      lim_{n→∞} \mid\frac{\frac{1}{n+1}}{\frac{1}{n}}\mid=lim_{n→∞} \mid\frac{n}{n+1}\mid


                                                             =lim_{n→∞} \mid\frac{1}{1+\frac{1}{n}}\mid

                                                (Click here for the limit step by step procedures)

                                                                                 =1=L


2. Given L, determine convergence

                                                         Since L = 1, the test is inconclusive.

You may be wondering what to do if the ratio series test is inconclusive. In the next couple of blog posts, we will be discussing other convergence tests that can be used when the ratio test is inconclusive. For more practice on the Ratio Series Test, check out Symbolab’s Practice.

Until next time,

Leah

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    Am shelny by name from UK i want to share my testimony to the world for the good help Dr Love0 granted to me. I have not been able to conceive for 5 years after marriage and this made my Husband Left me for another woman. I have been going through pains and hurt. I loved my husband so much that i cant stop thinking about him, a friend of mine who saw my condition and also know how i feel as woman she was looking for different way to help me, She called one day and said to me that there is this Great man somebody was talking about that he reunite lovers that i should contact him for help. That was how i contacted the great Dr Love0 and explained my problems to him and he told me not to worry that my man is going to come back within 24 hours and he will prepare some herbs for me that will make me pregnant, i never believed him until the 24 hours he promised, my husband called me and was apologizing to me that he needs me now in his life, i was so surprised because i felt it was a dream. So that was how my Husband came back to me March 15. Dr Love0 prepared his herbs for me and i used it a prescribed now am 2 months pregnant. What happened to me is like a dream and am telling people out there if Dr Love0 can solve my case i believe there is nobody Dr Love0 can not help. So if you need his help you contact him through:[email protected] or check his blogs: https://doctorlovespell0.blogspot.com/  or check website:  https://doctor0lovespell.wixsite.com/lovespell or visit his FB page: https://www.facebook.com/Love-Spell-Doctor0-107453264273672/ 
    Once again thanks be unto you the great Dr Love0 and God bless my friend who told me about Dr Love0.

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  55. I want to thank Dr Emu a very powerful spell caster who help me to bring my husband back to me, few month ago i have a serious problem with my husband, to the extend that he left the house, and he started dating another woman and he stayed with the woman, i tried all i can to bring him back, but all my effort was useless until the day my friend came to my house and i told her every thing that had happened between me and my husband, then she told me of a powerful spell caster who help her when she was in the same problem I then contact Dr Emu and told him every thing and he told me not to worry my self again that my husband will come back to me after he has cast a spell on him, i thought it was a joke, after he had finish casting the spell, he told me that he had just finish casting the spell, to my greatest surprise within 48 hours, my husband really came back begging me to forgive him, if you need his help you can contact him with via email: [email protected] or add him up on his whatsapp +2347012841542 is willing to help any body that need his help.
    Website: http://emutemple.website2.me/

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  56. I NEED AN URGENT 2020/2021 LOVE SPELL CASTER TO HELP BRING BACK EX HUSBAND AND RESTORE BROKEN MARRIAGE EMAIL [email protected]
    First of all i want to thank RUTH for the post she made on how DR Love0 helped her in bringing back her lover before NEW YEAR . At first when i saw the posting i was so happy and in the other hand so scared,That this might not be real, Then i decided to give it a try in which i contacted DR Love0 and told him how my lover left me for another lady for the past 3mouths and i have been lonely and depressed without him, So i told him if he has helped anyone called RUTH and he said yes, that was the lady he helped in bringing back her lover before NEW YEAR, I said good and i told him that if he can help me in bringing back my own lover, He laughed and said once i have contacted him that my problem will be solved. He said that my lover will be back to me within 24hours and do an unexpected thing for me. i said really, Truly when the 24hours was completed i got a text from someone saying am sorry then i decided to call the number i saw it was my lover smith voice. I was so happy he was begging me on phone, That he is ready to do anything that will make me happy in life, So i told him to come over which he did, As he was coming he came with a brand new Car as gift i was so happy and made me had access to his account to prove to me that he is not going to leave me for another lady, Am so happy today and am also thanking RUTH for posting this early. DR Love0 you are truly a man of your word. He can also solve any kind of problems in this world probles like WINNING LOTTERIES, GETTING OF JOB. BEAUTY SPELL CHILD BEARING. SPIRITUAL PROTECTION. JOB PROMOTION. BEAUTY SPELL HERBAL CARE money spell HIV spell. Friends you can contact DR Love0 on his private Email ([email protected]) or check his blogs: https://doctorlovespell0.blogspot.com/  or check website:  https://doctor0lovespell.wixsite.com/lovespell or visit his FB page: https://www.facebook.com/Love-Spell-Doctor0-107453264273672/

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  57. Thank you Dr Williams for what you have done for me am so greatful my lover is back to me and we are now living happily together. Dr Williams love spell is very powerful and effective and it does not have any side effect as he promised I decided to give Dr Williams a try when my lover left me for another he helped me to cast a love spell on my lover that brought him back to me what makes me excited the most is that my lover did not even know he is under a spell if you are passing through relationship love problem I advice you to contact Dr Williams to get your problem solve. It's very hard to loose a love one and I know how it feels so do not let somebody take away your lover from you. Contact Dr Williams today and get your problem solve and if you need his help below here is his email address.
    [email protected] com or you can also reach him on his WhatsApp +2348136785562

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