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 ?