Wednesday, March 18, 2020

Advanced Math - Series Convergence Calculator, Telescoping Series Test

Last blog post, we went over what an alternating series is and how to determine if it converges using the alternating series test. In this blog post, we will discuss another infinite series, the telescoping series, and how to determine if it converges using the telescoping series test.

If it isn’t clear right away, telescoping is synonymous with the word collapsing. A telescoping series is a series where almost all the terms cancel with the preceding or following term leaving just the initial and final terms, i.e. a series that can be collapsed into a few terms.

Let’s see what this looks like . . .

                                     ∑_{n=1}^∞\frac{1}{n(n+1)}= ∑_{n=1}^∞\frac{1}{n}-\frac{1}{n+1}

                                   = (1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+ ...+(\frac{1}{n}-\frac{1}{n+1})

                                                               =1-\frac{1}{n+1}

As you can see, we are able to cancel out all terms except the first and last.

Now that we’ve discussed what a telescoping series is, let’s go over the telescoping series test.

Telescoping Series Test:

For a finite upper boundary, ∑_{n=k}^N(a_{n+1}-a_n )=a_{N+1 }-a_k
For an infinite upper boundary, if a_n→0*, then ∑_{n=k}^∞(a_{n+1}-a_n )= -a_k
*If a_n doesn’t converge to 0, then the series diverges.

In regards to infinite series, we will focus on the infinite upper boundary scenario. In order to use this test, you will need to manipulate the series formula to equal a_{n+1}-a_n where you can easily identify what a_{n+1} and a_n are. Also, please note that if you are able to manipulate the series in this form, you can confirm that you have a telescoping series. With practice, this will come more naturally.

Let’s see some examples to better understand.

First example (click here):

                                                         ∑_{n=1}^∞\frac{5}{n}-\frac{5}{n+1}


1. Convert the series into the form a_{n+1}-a_n

                                                        \frac{5}{n}-\frac{5}{n+1}= -\frac{5}{n+1}-(-\frac{5}{n})

                                                               a_{n+1}=-\frac{5}{n+1}

                                                                      a_n=-\frac{5}{n}

2. Determine if a_n→0

                                                               a_n=-\frac{5}{n}= -5(\frac{1}{n})

                                              Since \frac{1}{n} converges to 0, -\frac{5}{n} converges to 0.

3. Calculate -a_k

                                                                         k=1

                                                                     a_n=-\frac{5}{n}

                                                             -a_k=-(-\frac{5}{1})=5
             The series converges to 5.

Next example (click here)

                                                          ∑_{n=1}^∞\frac{6}{(n+1)(n+2)}

1. Convert the series into the form a_{n+1}-a_n

                                   ∑_{n=1}^∞\frac{6}{(n+1)(n+2)}= 6∙∑_{n=1}^∞\frac{1}{(n+1)(n+2)}

                                                 \frac{1}{(n+1)(n+2)}= -(\frac{1}{n+2})-(-\frac{1}{n+1})

                                                                 a_{n+1}=-\frac{1}{n+2}

                                                                     a_n=-\frac{1}{n+1}


2. Determine if a_n→0

                                                                     a_n=-\frac{1}{n+1}

                                                                    -\frac{1}{n+1}  →0

3. Calculate -a_k

                                                                            k=1

                                                                     a_n=-\frac{1}{n+1}

                                                           -a_k=-(-\frac{1}{1+1})=\frac{1}{2}


                                                   6∙∑_{n=1}^∞\frac{1}{(n+1)(n+2)} =6∙\frac{1}{2}=3

             The series converges to 3.

Last example (click here):

                                                                ∑_{n=1}^∞\frac{1}{4n^2-1}

1. Convert the series into the form a_{n+1}-a_n

                                                \frac{1}{4n^2-1}=-(\frac{1}{2(2n+1)} )-(-\frac{1}{2(2n-1)})

                                                                a_{n+1}= -(\frac{1}{2(2n+1)} )

                                                                    a_n=-\frac{1}{2(2n-1)}

2. Determine if a_n→0

                                                              a_n=-\frac{1}{2(2n-1)} =-\frac{1}{4n-1}

                                                                      -\frac{1}{4n-1}  →0


3. Calculate -a_k

                                                                               k=1

                                                                     a_n=-\frac{1}{2(2n-1)}

                                                           -a_k=-(-\frac{1}{2(2∙1-1)} )=\frac{1}{2}

              The series converges to \frac{1}{2}.

The trickiest part of this is manipulating the series formula into a_{n+1}-a_n. Once you’re able to do this, the rest should be pretty simple. The key thing to remember about a telescoping series is that all the terms will cancel out, except the first and last term.

For more help on telescoping series, check out Symbolab’s Practice. Next blog post, I’ll go over the convergence test for a radio series.

Until next time,

Leah

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  2. Melbet
    Kaçak iddaa siteleri arasında olan Melbet zaman zaman engelleniyor. Bu duruma karşılık önlemlerini aldığı için işiniz uzun sürmez. Yani son adresini gözler önüne serer ve katılmanız uzun sürmez. Katılımcılara:
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    Melbet Adresi Değişti Mi?
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    Yukarıda da belirttiğimiz üzere hem adres duyuruları hem de linkleri bulunuyor. Bu sayede bilgi sahibi olmanız ve katılmanız uzun sürmez. Katılımcıları aynı tasarım özellikleri ve hizmetleri karşılar. Şöyle ki:
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    melbet



    —————————————————————————————————————


    Melbet Yeni Giriş Adresi
    Canlı bahislerden TV oyunlarına kadar çeşitli hizmetlerin yer aldığı Melbet adres değişikliğine gitti. Katılmak isteyenler yeni adresi öğrenmeli ve erişim sağlamalı. Ancak bu yolla içerisinde neler var göz atmanız ve faaliyet göstermeniz mümkün olacaktır.
    Melbet Yeni Adresi Nedir?
    Melbet giriş adresini BTK engellerinden dolayı değiştiren siteler arasında. Bu durumu dikkate alan firma anında domain değişikliğine gider. Ve mevcut hizmetlerini sunmaya devam eder. Bunlar:
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    Sadece web siteye ilişkin adres değişikliğine gidilmemiştir. Haricinde mobil sürüm için de güncel adresini sergilemiştir. Giriş yapmak isteyenler mobil için duyurulan son adresi kullanmak zorunda. Neticesinde:
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    melbet



    ReplyDelete
  3. Redfoxbet Yeni Giriş Adresi
    Canlı destek hattı ile kullanıcılarını karşılayan siteler arasında Redfoxbet’te yerini almıştır. Hatta üye olmayanlar dahi destek hattına katılabilir ve sorularını iletebilir. Bunun için ilk etapta:
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    * Mobil için de yine son adresi kullanmanız gerekmekte.
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    Redfoxbet Neden Engellendi?
    Adres engelleri kaçak siteler üzerinden gerçekleşir. Redfoxbet giriş adresi bloke edilen platformlar arasında. Çünkü Türkiye yasalarına uygun herhangi bir lisansı mevcut değil.
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    Redfoxbet’e Nasıl Giriş Yapılır?
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    * Giriş linklerine tıklayarak siteyi açabilirsiniz.
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    redfoxbet

    ReplyDelete
  4. Tipobet Yeni Giriş Adresi
    Uzun yıllardır sektörde olan Tipobet, adres paylaşımlarına dikkat eder. Ülkemizden katılımcıların bu sayede sorun yaşaması mümkün olmaz. Öyle ki BTK engellerinden dolayı:
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    * Sadece engelsiz adresten giriş yapmanıza izin verilir.
    Tipobet engelsiz adrese ilişkin duyurularını sergilemiştir. Bununla sınırlı kalmayıp giriş linklerini de sunması dikkatlerden kaçmıyor. Yani link tıklaması ile beraber Türkçe olan siteye erişim sağlanabilir.
    Tipobet Resmi Adresi Ne Oldu?
    Tipobet365 sitesi nasıl merak edenler başta erişim sağlamalı ve incelemeli. Türkçe olan siteyi kısa sürede analiz etmeniz mümkün olacaktır. Bu anlamda canlı destek hattını da sunması işinizi kolay hale getirecektir.
    Resmi adrese ilişkin paylaşımlarına devam ediyor. Giriş yapanlar dolandırıcı sitelerden uzak durmak için başta resmi adresi kontrol etmeli. Bu sayede herhangi bir sorun yaşamadan giriş yapmak ve harekete geçmek mümkün olacaktır.
    Tipobet Mobil Adresi Değişti Mi?
    Telefonlardan erişime uygun olan site, yine güncel adresi kullanmanızı bekliyor. Çünkü mobil için de BTK engellerine yer verilmiştir. Bu duruma karşılık:
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    * Telefonlardan adres araması yapanlar mobil sürümü açacaktır.
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    tipobet

    ReplyDelete
  5. Vdcasino Yeni Giriş Adresi
    Sadece casino oyunlarına yer vermeyen Vdcasino, ilk etapta maç yayını sunar. Kesintisiz ve ücretsiz olarak canlı tv sunmaya devam ediyor. Buna bağlı olarak spor ile canlı bahisleri de sunmuştur. Hepsine:
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    Ödemeler beraberinde bonus kazanmanıza olanak sağlar. Örneğin çevrim şartsız olarak yatırım fırsatlarını talep edebilirsiniz.
    Vdcasino Giriş Adresi Değişti Mi?
    Bilgi Teknolojileri ve İletişim Kurumu engellerine maruz kalan siteler arasında Vdcasino’da yerini almıştır. Engeller adres üzerinden gerçekleştiği için yine domain değişikliğine başvurulmuştur.
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    Vdcasino Resmi Adresi Nedir?
    Resmi adrese ilişkin paylaşımlarına devam ediyor. Özellikle de sosyal medya hesaplarından bu konuya değinmiştir. Haricinde duyurular ve linkler devreye girmiştir. Böylece üyeler:
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    vdcasino

    ReplyDelete
  6. Bahsegel Güncel Giriş Adresi
    Yeni adres bildirimlerine açık olan Bahsegel, bu şekilde üyelerini bilgilendiriyor. Üyeler zaman kaybetmeden giriş yapabiliyor. Aynı hesap verileri üzerinden katılmaları ve faaliyet göstermeleri sağlanıyor. Şöyle ki:
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    Neticesinde de bahislere ya da casino alanlarına katılabilirsiniz. En yeni promosyonları da bu yolla talep etmekte zaman kaybetmezsiniz.
    Bahsegel Girişi Nasıl?
    Bahsegel giriş adresine ilişkin duyurularını ve linklerini esirgemiyor. Özel bildirimlere yer vermesi de işinizi kolay hale getirir. Bu sayede erişim sağlamanız uzun sürmez ve anında faaliyet gösterirsiniz.
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    * Canlı Casino
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    Bahsegel Türkiye Giriş Adresi
    Türkiye’den yasal izinleri bulunmadığı için engellenen Bahsegel, kaldığı yerden hizmetlerini sunmaya devam ediyor. Yine Türkiye’den kullanıcılarına göre tasarlanan site, Curacao lisansı eşliğinde bahislerini sergiliyor.
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    bahsegel


    ReplyDelete
  7. Bahsegel  Giriş Adresi
    Adresi bloke edildiği için bu alanda değişiklik yapan Bahsegel, Türkiye’den bahis severleri karşılamaya devam ediyor. Bunun için Türkçe hizmetlere ev sahipliği yapmakta. İlk etapta giriş yapanlara sergilenen hizmetleri de:
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    * Canlı Bahis
    * Casino
    * Canlı Casino
    * Canlı Oyunlar
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    Bahsegel giriş adresini değiştirmiş ve sektöre yeniden adım atmıştır. Kapatılmasına yönelik herhangi bir çalışma gerçekleşmemiştir. BTK tarafından sadece adres engeline yer verilmiştir.
    Neticesinde güncel adreslerini duyuran site, üyelerini karşılamaya devam ediyor. Kullanıcılar direk olarak oturum açabilir ve bu yolla hizmet alabilir. Yukarıda sıraladığımız bahis ya da casino seçeneklerinden tercihte bulunanların işi uzun sürmez.
    Bahsegel Mobil Adresi Ne Oldu?
    BTK engellerinden dolayı eski adresi kullanarak mobil açmanız mümkün değil. Yine bu aşamada güncel adresi nedir dikkate almak gerekir. Ancak bu yolla açılacak olan mobil siteden:
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    bahsegel

    ReplyDelete
  8. Melbet
    Koyu Renkler ve sari rengin hakim olduğu bir renk paleti ile Göze Hitap eden bahis firmasi Melbet, Continental Solutions Ltd Curacao Lisansı ile yetkilidir ve güvenli oyun temeliyle hizmet verir.
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    melbet


    ReplyDelete
  9. Sultanbet
    Yeşil ve siyah tasarımıyla mükemmel bir atmosferi yansıtan Sultanbet, Continental Solutions Ltd Curacao Lisansı ile yetkilidir ve güvenli oyun temeliyle hizmet verir.
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    sultanbet

    ReplyDelete
  10. Süperbahis
    Tarihin en eski bahis sitesinde yüksek kazançlar sizi bekliyor! Süperbahis Curacao lisans yetkileriyle donatılmış mavi kırmızı tasarımı ve 5536/JAZ lisans numarasıyla yıllara meydan okumaya devam ediyor. Bahis denilince akla ilk gelen güvenilir site olmanın verdiği güvenle hizmet veriyor.
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    superbahis

    ReplyDelete
  11. Portbet
    Siyah, turuncu ve beyaz renklerin hâkim olduğu tasarımıyla göze çarpan Portbet, 8048/JAZ Curacao lisansına sahiptir. Firmaya ilk üyeliklerini gerçekleştiren bahisçiler, ilk yatırımlarına özel çevrim şartsız 250 TL hoş geldin bonusu ve 500 TL çevrim şartlı hoş geldin bonusu vermektedir.
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    portbet



    ReplyDelete
  12. Marsbahis
    Online canlı bahis ve casino sitesi Marsbahis, 129948 kayıt numaralı Curacao lisansına sahip bir firmadır. Firmaya üyelik işlemlerini gerçekleştiren bahisçiler, Marsbahis giriş adresi www.marsbahis.com olarak güncellenmiştir. Bahisçiler, üyelik işlemleri için, kullanıcı adı ve şifresini girmesi yeterlidir.
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    Marsbahis Yatırım Yöntemleri Nelerdir?
    Türkçe ve İngilizce dil desteği sunan Marsbahis, sunduğu para yatırma yöntemleriyle de üyelerin dikkatini çekmektedir. Firmanın para yatırma yöntemleri arasında;
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  92. BEST POWERFUL AND CHEAP LOVE SPELL CASTER DR JUMBA

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