Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

Key Questions

• What do you do if the Alternating Series Test fails?

  In most cases, an alternation series #sum_{n=0}^infty(-1)^nb_n# fails Alternating Series Test by violating #lim_{n to infty}b_n=0#. If that is the case, you may conclude that the series diverges by Divergence (Nth Term) Test.

  I hope that this was helpful.

  Wataru · 1 · Sep 28 2014
• How do I determine if the alternating series #sum_(n=1)^oo(-1)^n/sqrt(3n+1)# is convergent?

  Alternating Series Test

  An alternating series #sum_{n=1}^infty(-1)^n b_n#, #b_n ge 0# converges if both of the following conditions hold.

  #{(b_n ge b_{n+1} " for all " n ge N),(lim_{n to infty}b_n=0):}#

  ---

  Let us look at the posted alternating series.

  In this series, #b_n=1/sqrt{3n+1}#.

  #b_n=1/sqrt{3n+1} ge 1/sqrt{3(n+1)+1}=b_{n+1}# for all #n ge 1#.

  and

  #lim_{n to infty}b_n=lim_{n to infty}1/sqrt{3n+1}=1/infty=0#

  Hence, we conclude that the series converges by Alternating Series Test.

  ---

  I hope that this was helpful.

  Wataru · 1 · Oct 18 2014
• What is the Alternating Series Test of convergence?

  Alternating Series Test states that an alternating series of the form
  #sum_{n=1}^infty (-1)^nb_n#, where #b_n ge0#,
  converges if the following two conditions are satisfied:
  1. #b_n ge b_{n+1}# for all #n ge N#, where #N# is some natural number.
  2. #lim_{n to infty}b_n=0#

  Let us look at the alternating harmonic series #sum_{n=1}^infty (-1)^{n-1}1/n#.
  In this series, #b_n=1/n#. Let us check the two conditions.
  1. #1/n ge 1/{n+1}# for all #n ge 1#
  2. #lim_{n to infty}1/n=0#

  Hence, we conclude that the alternating harmonic series converges.

  Wataru · · Sep 12 2014

• What is the Alternating Series Test of convergence?
• Can the Alternating Series Test prove divergence?
• Does the Alternating Series Test determine absolute convergence?
• How do you use the Alternating Series Test?
• What do you do if the Alternating Series Test fails?
• What is an example of an alternating series?
• How do I find the sum of the series: 4+5+6+8+9+10+12+13+14+⋯+168+169+170. since D is changing from +1, +1 to +2 ?
• How do you determine the convergence or divergence of #sum_(n=1)^(oo) (-1)^(n+1)/n#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n+1)n)/(2n-1)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n+1))/(2n-1)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n))/(ln(n+1))# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n)n^2)/(n^2+1)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n)n)/(n^2+1)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n))/(sqrtn)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n+1)n^2)/(n^2+5)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^(n+1)ln(n+1))/((n+1))# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma sin(((2n-1)pi)/2)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma 1/nsin(((2n-1)pi)/2)# from #[1,oo)#?
• How do you determine the convergence or divergence of #sum_(n=1)^(oo) cosnpi#?
• How do you determine the convergence or divergence of #Sigma 1/ncosnpi# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma(-1)^n/(n!)# from #[1,oo)#?
• How do you determine the convergence or divergence of #sum_(n=1)^oo (-1)^n/((2n-1)!)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^nsqrtn)/root3n# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma ((-1)^n n!)/(1*3*5***(2n-1)# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma (-1)^(n+1)(1*3*5***(2n-1))/(1*4*7***(3n-2))# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma (-1)^(n+1)cschn# from #[1,oo)#?
• How do you determine the convergence or divergence of #Sigma (-1)^(n+1)sechn# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (-1)^(n+1)/(n+1)^2# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (-1)^(n+1)/(n+1)# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (-1)^(n+1)/sqrtn# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n+1)n^2)/(n+1)^2# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n))/(lnn)# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n+1))/(n^1.5)# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n))/((2n+1)!)# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n))/(sqrt(n+4))# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (cos(npi))/(n+1)# from #[1,oo)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #sum_(n=1)^oo (-1)^(n+1)arctan(n)#?
• How do you determine if the series the converges conditionally, absolutely or diverges given #sum_(n=1)^oo (cos(npi))/n^2#?
• How do you test the alternating series #Sigma (-1)^nsqrtn# from n is #[1,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^(n+1)/sqrtn# from n is #[1,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^(n+1)n/(10n+5)# from n is #[1,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^nsqrtn/(n+1)# from n is #[1,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^n/lnn# from n is #[2,oo)# for convergence?
• How do you test the alternating series #Sigma (n(-1)^(n+1))/lnn# from n is #[2,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^n# from n is #[1,oo)# for convergence?
• How do you test the alternating series #Sigma ((-1)^(n+1)2^n)/(n!)# from n is #[0,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^n/(ln(lnn))# from n is #[3,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^n/rootn n# from n is #[2,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^(n+1)(1-1/n)# from n is #[1,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^n(sqrt(n+1)-sqrtn)# from n is #[1,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^n(2^n+1)/(3^n-2)# from n is #[0,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^n(2^(n-2)+1)/(2^(n+3)+5)# from n is #[0,oo)# for convergence?
• How do you test the alternating series #Sigma (-1)^(n+1)(1+1/n)# from n is #[1,oo)# for convergence?
• For what values of #r# does the sequence #a_n = (nr)^n# converge ?