Ratio Test for Convergence of an Infinite Series

Key Questions

• How do you know when to use the Ratio Test for convergence?

  It is not always clear-cut, but if a series contains exponential functions or/and factorials, then Ratio Test is probably a good way to go.

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  I hope that this was helpful.

  Wataru · 4 · Oct 19 2014
• How do you use the Ratio Test on the series #sum_(n=1)^oo(-10)^n/(4^(2n+1)(n+1))# ?

  By Ratio Test, the posted series converges absolutely.

  By Ratio Test:

  #lim_{n to infty}|a_{n+1}/a_n|=lim_{n to infty}|{(-10)^{n+1}}/{4^{2n+3}(n+2)}cdot{4^{2n+1}(n+1)}/{(-10)^n}|#

  By canceling out common factors:

  #=lim_{n to infty}|{-10(n+1)}/{4^2(n+2)}|#

  since #|{-10}/4^2|=5/8#, we have:

  #=5/8lim_{n to infty}(n+1)/(n+2)#

  by dividing the numerator and the denominator by #n#,

  #=5/8 lim_{n to infty}{1+1/n}/{1+2/n}=5/8cdot 1=5/8<1#

  Hence, #sum_{n=1}^{infty}{(-10)^n}/{4^{2n+1}(n+1)}# is absolutely convergent.

  Wataru · · Sep 4 2014

• How do you know when to use the Ratio Test for convergence?
• How do you use the Ratio Test on the series #sum_(n=1)^oon^n/(n!)# ?
• How do you use the Ratio Test on the series #sum_(n=1)^oo(n!)/(100^n)# ?
• How do you use the Ratio Test on the series #sum_(n=1)^oo(-10)^n/(4^(2n+1)(n+1))# ?
• How do you use the Ratio Test on the series #sum_(n=1)^oo9^n/n# ?
• How do you use the ratio test to test the convergence of the series #∑ (-5)^(n+1)n / 2^n# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ 3^n/(4n³+5)# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #sum_(n=1)^oo((x+1)^n) / (n!)# ?
• How do you use the ratio test to test the convergence of the series #∑3^k/((k+1)!)# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑(2k)!/k^(2k) # from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑(4^n) /( 3^n + 1)# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑(n!)/(n^n)# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑(5^k+k)/(k!+3)# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑(x^(n))/(9^(n))# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #sum_(n=1)^oo (n!)/((2n+1)!)#?
• How do you use the ratio test to test the convergence of the series #∑ (n!)^2 / (kn)!# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ (3/4)^n# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ (n+1)/(3^n)# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ [n(n!)^2]/(2n+1)!# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ 11^n/((n+1)(7^(2n+1)))# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ (2n^2)/(n!) # from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ (8^n)/(n!)# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑ ((4n+3)^n) / ((n+7)^(2n))# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑( (3n)!) / (n!)^3# from n=1 to infinity?
• How do you use the ratio test to test the convergence of the series #∑k/(3+k^2) # from k=1 to infinity?
• How do you apply the ratio test to determine if #sum_(n=1)^oo 3^n# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma 1/sqrtn# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma 2^n/(n!)# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma 1/n^3# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma (5^n)/(6^n-5^n)# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma (n!)/n^n# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma n^n/((2n)!)# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma (3^n(n!))/(n^n)# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #sum_(n=1)^oo (e^n(n!))/n^n# is convergent or divergent?
• How do you apply the ratio test to determine if #Sigma (4^n(n!)^2)/((2n)!)# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma (3^n(n!)^2)/((2n)!)# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma 1/(lnn)^n# from #n=[2,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #sum_(n=2)^oo 10^n/(lnn)^n# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma 1/(ln(lnn))^n# from #n=[3,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma (n!)/(1*3*5* * *(2n-1))# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma (1*3*5* * * (2n-1))/(n!)^2# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #sum (1*3*5* * * (2n-1))/(1*4*7* * * (3n-2))# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma (n!)^3/((3n)!)# from #n=[1,oo)# is convergent to divergent?
• How do you apply the ratio test to determine if #Sigma 1/sqrt(n!)# from #n=[1,oo)# is convergent to divergent?
• How do you test the series #Sigma (2^n(n!))/n^n# from #n=[1,oo)# by the ratio test?
• How do you test the series #Sigma (n^n)/(lnn)^n# from #n=[2,oo)# by the ratio test?
• How do you test the series #Sigma (100^n(n!)^3)/((3n)!)# from #n=[1,oo)# by the ratio test?
• #sum_(n=1)^oo sin(n)/(n!)# How would i find if it converges or diverges?
• The series #sum_(n=1)^oo x^n/10^n # converges for #|x| lt beta#, find #beta#?