Integral Test for Convergence of an Infinite Series

Key Questions

• How do you know when to use the integral test for an infinite series?

  Well, I would avoid using the integral test since evaluating an integral can be very difficult. If nothing else works and you know how to evaluate the integral, then go for it.

  Wataru · · Sep 11 2014
• How do you use the Integral test on the infinite series #sum_(n=1)^oo1/n^5# ?

  By Integral Test,

  #sum_{n=1}^infty 1/n^5# converges.

  Let us look at some details.

  Let us evaluate the corresponding improper integral.

  #int_1^infty 1/x^5 dx#

  #=lim_{t to infty}int_1^tx^{-5} dx#

  #=lim_{t to infty}[x^{-4}/-4]_1^t#

  #=-1/4lim_{t to infty}[1/x^4]_1^t#

  #=-1/4 lim_{t to infty}[1/t^4-1]#

  #=-1/4(0-1)=1/4#

  Since the integral

  #int_1^infty 1/x^5 dx#

  converges to #1/4#,

  #sum_{n=1}^infty 1/n^5#

  also converges by Integral Test.

  Wataru · · Sep 24 2014
• What is the Integral Test for Convergence of an Infinite Series?

  Integral Test

  If #f# is a function such that #f(n)=a_n#, then

  #sum_{n=1}^inftya_n# and #int_1^infty f(x)dx# converge or diverge together.

  ---

  I hope that this was helpful.

  Wataru · · Oct 16 2014

• What is the Integral Test for Convergence of an Infinite Series?
• How do you know when to use the integral test for an infinite series?
• How do you use the Integral test on the infinite series #sum_(n=1)^oo1/root5(n)# ?
• How do you use the Integral test on the infinite series #sum_(n=1)^oo1/n^5# ?
• How do you use the Integral test on the infinite series #sum_(n=1)^oo1/(2n+1)^3# ?
• How do you use the Integral test on the infinite series #sum_(n=1)^oo1/sqrt(n+4)# ?
• How do you determine if the series #ln(1/2) + ln(1/3) + ln(3/4) + ... +ln[k/(k + 1)] + ....# converges?
• How do you know #{-1,1,-1,1,-1,1,...}# converges or diverges?
• Using the integral test, how do you show whether # (1 + (1/x))^x# diverges or converges?
• Using the integral test, how do you show whether #sum 1/(n^2+1)# diverges or converges from n=1 to infinity?
• How do you use the Integral Test to determine convergence or divergence of the series: #sum n e^-n# from n=1 to infinity?
• Using the integral test, how do you show whether #(1/sqrt (n+1))# diverges or converges?
• Using the integral test, how do you show whether #sum1/[(n^2)+4)# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #1/(2n+3)# diverges or converges?
• Using the integral test, how do you show whether #n/(n^2+1)# diverges or converges?
• Using the integral test, how do you show whether #sum 1 / [sqrt(n) * (sqrt(n) + 1)]# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum 1/n^2# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum (1/e^k)# diverges or converges?
• How do you determine convergence or divergence for the summation of #n*e^(-n/2)# using the integral test how do you answer?
• Use the Integral Test to determine whether the series is convergent or divergent given #sum 1 / n^5# from n=1 to infinity?
• How do you use the integral test to determine whether the following series converge of diverge #sum n/((n^2+1)^2)# from n=1 to infinity? Thanks for the help !!! I have no idea on how to do these questions?
• Using the integral test, how do you show whether #sum 1/(nln(3n))# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum 1/(n*(n))# diverges or converges from n=1 to infinity?
• Does #sum_{n=2} 1 / (1 + n ( Ln(n) )^2)# converges or diverges from n=2 to infinity?
• Using the integral test, how do you show whether #sum ln(n)/(n)^2# diverges or converges from n=1 to infinity?
• How do you use the integral test to find whether the following series converges or diverges #sum( 1/(n*ln(n)^0.5) )#?
• How do you use the integral test to determine the convergence or divergence of the series from n=1 to infinity for #(arctan n) / (n^2 + 1)#?
• Using the integral test, how do you show whether #sum 1/n^3# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum (1/n^2)cos(1/n) # diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum 1/(n(lnn)^2) # diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum (n + 2) / (n + 1)# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum1/(n^2 - 1)# diverges or converges from n=4 to infinity?
• Using the integral test, how do you show whether #sum1 / (n (log n)^p ) # diverges or converges from n=3 to infinity?
• Using the integral test, how do you show whether #sum 3/(n sqrt(ln(n)))# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum 1/sqrt(n+1)# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum 1/((2n+1)^2)# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum 1/sqrt(2x-5)# diverges or converges from n=1 to infinity?
• Using the integral test, how do you show whether #sum 1 / (n^2 + 1)# diverges or converges from n=2 to infinity?
• How do you show that #sum(n-1)/(n*4^n)# is convergent using the Comparison Test or Integral Test?
• How do you show whether the improper integral #int lim (lnx) / x dx# converges or diverges from 1 to infinity?
• How do you show whether the improper integral #int dx / [(lnx)^2]# converges or diverges from e to infinity?
• How do you show whether the improper integral #int dx / (x^2 + sinx)# converges or diverges from 2 to infinity?
• How do you show whether the improper integral #int (1/3x-6) dx# converges or diverges from negative infinity to zero?
• How do you show whether the improper integral #int (x^2)/(9+x^6) dx# converges or diverges from negative infinity to infinity?
• Question #74f47
• Does #sum_1^oo [3/n(n+4)] - [(2^(2n)/7^n+1)]# converge?
• How do you determine if the improper integral converges or diverges #int_0^oo 1/ (x-2)^2 dx #?
• How do you determine if the improper integral converges or diverges #e^(-2t) dt # from negative infinity to -1?
• How do you determine if the improper integral converges or diverges #int (1 / (u^2 + 3))du # from 0 to infinity?
• How do you determine if the improper integral converges or diverges #int ln(sin(x))# from 0 to pi/2?
• How do you determine if the improper integral converges or diverges #int 1/x*(lnx)^p dx# from 2 to infinity?
• How do you determine if the improper integral converges or diverges #int (x^3 + x)/((x^4 + 2x^2 + 2)^(1/2))dx# from 1 to infinity?
• How do you determine if the improper integral converges or diverges #int ln(x)dx# from 0 to 2?
• How do you determine if the improper integral converges or diverges #int dx/(2x-1)# from 0 to infinity?
• How do you determine if the improper integral converges or diverges #int xe^-x dx # from 0 to infinity?
• How do you determine if the improper integral converges or diverges #int 3/(7x^2 +4) # from negative infinity to infinity?
• How do you determine if the improper integral converges or diverges #int 1 / [sqrt x] # from 0 to infinity?
• How do you determine if the improper integral converges or diverges #int dx/((3x-2)^6) # from 2 to infinity?
• How do you determine if the improper integral converges or diverges #int 8dx/(x^(2)+1)# from 1 to infinity?
• How do you determine if the improper integral converges or diverges #int 5x^(2)e^(-x^(3))# from 1 to infinity?
• How do you determine if the improper integral converges or diverges #int [e^(1/x)] / [x^3]# from 0 to 1?
• How do you determine if the improper integral converges or diverges #int [(x^3)( e^(-x^4) )] dx# from negative infinity to infinity?
• How do you determine if the improper integral converges or diverges #int sec^2 x dx# from negative 0 to pi?
• How do you determine if the improper integral converges or diverges #int [ (x arctan x) / (1+x^2)^2 ] dx# from negative 0 to infinity?
• How do you determine if the improper integral converges or diverges #int (1/(3x)-6) dx# from negative infinity to 0?
• How do you determine if the improper integral converges or diverges #intx^2 e^-x dx# from 0 to infinity?
• How do you determine if the improper integral converges or diverges #int (e^x)/(e^2x + 1) dx# from 0 to infinity?
• What is #int x*e^(x^2) dx# ?
• Question #ba17d
• Question #720c6
• Question #cdd90
• Question #cddb4
• Question #168a1
• Question #721f3
• Question #ca2da
• Question #dc166
• #lim_(n->oo)(1/n((n+1)(n+2)(n+3)cdots(2n))^(1/n))?#
• #lim_(n->oo)((sqrt(1)+sqrt(2)+sqrt(3)+cdots+sqrt(n))/(n sqrt(n)))# ?
• #lim_(n->oo)(1/(1 xx 2)+1/(2 xx 3)+1/(3 xx4) + cdots + 1/(n(n+1)))#?
• Prove that for #n > 1# we have #1 xx 3 xx 5 xx 7 xx cdots xx(2n-1) < n^n#?
• Question #eca11
• Question #084f4
• Lim n approaches infinity# 6/n((2n)/3 + (5n(n+1))/(2n) - (4n(n+1)(2n+1))/(6n^2))=#?
• How do you use the integral test to determine if #Sigma e^-n# from #[1,oo)# is convergent or divergent?
• How do you use the integral test to determine if #Sigma e^(-n/2)# from #[1,oo)# is convergent or divergent?
• How do you use the integral test to determine if #1/2+1/5+1/10+1/17+1/26+...# is convergent or divergent?
• How do you use the integral test to determine if #1/3+1/5+1/7+1/9+1/11+...# is convergent or divergent?
• How do you use the integral test to determine if #ln2/2+ln3/3+ln4/4+ln5/5+ln6/6+...# is convergent or divergent?
• How do you use the integral test to determine if #ln2/sqrt2+ln3/sqrt3+ln4/sqrt4+ln5/sqrt5+ln6/sqrt6+...# is convergent or divergent?
• How do you use the integral test to determine if #1/(sqrt1(sqrt1+1))+1/(sqrt2(sqrt2+1))+1/(sqrt3(sqrt3+1))+...1/(sqrtn(sqrtn+1))+...# is convergent or divergent?
• How do you use the integral test to determine if #1/4+2/7+3/12+...+n/(n^2+3)+...# is convergent or divergent?
• How do you use the integral test to determine if #Sigma1/sqrt(n+1)# from #[1,oo)# is convergent or divergent?
• How do you use the integral test to determine if #Sigma lnn/n^3# from #[2,oo)# is convergent or divergent?
• How do you use the integral test to determine if #Sigma lnn/n^2# from #[1,oo)# is convergent or divergent?
• How do you use the integral test to determine if #sum_(n=2)^oo 1/(nsqrtlnn)# from #[2,oo)# is convergent or divergent?
• How do you use the integral test to determine if #Sigma arctann/(n^2+1)# from #[1,oo)# is convergent or divergent?
• How do you use the integral test to determine if # sum_(n=3)^(oo) 1/(nlnnln(lnn))# is convergent or divergent?
• How do you use the integral test to determine if #Sigma n/(n^4+1)# from #[1,oo)# is convergent or divergent?
• How do you use the integral test to determine if #Sigma n^ke^-n# from #[1,oo)# where k is an integer is convergent or divergent?
• Why does the integral test not apply to #Sigma (-1)^n/n# from #[1,oo)#?
• Why does the integral test not apply to #Sigma (2+sinn)/n# from #[1,oo)#?
• Why does the integral test not apply to #Sigma (sinn/n)^2# from #[1,oo)#?
• How do you use the integral test to determine the convergence or divergence of #Sigma 1/n^3# from #[1,oo)#?
• How do you use the integral test to determine the convergence or divergence of #Sigma 1/n^(1/3)# from #[1,oo)#?
• How do you use the integral test to determine the convergence or divergence of #Sigma 1/sqrtn# from #[1,oo)#?
• How do you use the integral test to determine the convergence or divergence of #Sigma 1/n^2# from #[1,oo)#?
• How do you use the integral test to determine the convergence or divergence of #Sigma 1/root5(n)# from #[1,oo)#?
• How do you use the integral test to determine the convergence or divergence of #sum_(n=1)^(infty) 3/(n^(5/3))#?
• How do you use the integral test to determine the convergence or divergence of #1+1/sqrt2+1/sqrt3+1/sqrt4+...#?
• How do you use the integral test to determine the convergence or divergence of #1+1/4+1/9+1/16+1/25+...#?
• How do you use the integral test to determine the convergence or divergence of #1+1/(2sqrt2)+1/(3sqrt3)+1/(4sqrt4)+1/(5sqrt5)+...#?
• How do you use the integral test to determine the convergence or divergence of #1+1/root3(4)+1/root3(9)+1/root3(16)+1/root3(25)+...#?
• How do you find the positive values of p for which #Sigma (1/n(lnn)^p)# from #[2,oo)# converges?
• How do you find the positive values of p for which #Sigma lnn/n^p# from #[2,oo)# converges?
• How do you find the positive values of p for which #Sigma n/(1+n^2)^p# from #[2,oo)# converges?
• How do you find the positive values of p for which #Sigma n(1+n^2)^p# from #[2,oo)# converges?
• How do you show whether #sum_(n=2)^oo 1/ln^3(n)# converges or diverges?
• Question #94f10
• #lim_(n->oo)(1^alpha+2^alpha+cdots+n^alpha)/n^(alpha+1) =#?
• How do you use the integral test to determine whether #int dx/lnx# converges or diverges from #[2,oo)#?
• How do you use the integral test to determine whether #int dx/(x+lnx)# converges or diverges from #[2,oo)#?
• How do you use the integral test to determine whether #int (x+1)/(x^3+x^2+1)# converges or diverges from #[1,oo)#?
• How do you use the integral test to determine whether #int lnx/(xsqrtx)# converges or diverges from #[3,oo)#?
• How do you use the integral test to determine whether #int e^(-x^2)# converges or diverges from #[0,oo)#?
• How do you use the integral test to determine whether #int x^-x# converges or diverges from #[1,oo)#?
• The integral #int_0^a (sin^2x)/x^(5/2)dx# converges or diverges ?
• Does #int_1^oo1/root(3)(x+1)dx# converge or diverge? If it converges, what is the integral?
• How do you prove that #sum_(n=1)^oo (n^(1/n)-1)# diverges?
• Question #70731
• #int_0^oo 17 e^(-10s)ds = # ?
• Question #f3a16
• Does the function converge or diverge #1/((x-1)(x^2+1)) #on the bound [2,infinity]?
• Question #b5a4d