Direct Comparison Test for Convergence of an Infinite Series

Key Questions

• How do you use the direct Comparison test on the infinite series `sum_(n=1)^oo9^n/(3+10^n)` ?

  By Comparison Test, we can conclude that the series
  `sum_{n=1}^infty{9^n}/{3+10^n}` converges.

  Let us look at some details.
  For all `n geq 1`,
  `9^n/{3+10^n} leq 9^n/10^n=(9/10)^n`

  By Geometric Series Test,
  `sum_{n=1}^infty(9/10)^n` converges since `|r|=9/10<1`

  Hence, by Comparison Test,
  `sum_{n=1}^infty{9^n}/{3+10^n}` also converges.

  Wataru · · Sep 11 2014
• How do you use the direct comparison test for improper integrals?

  If an improper integral diverges, then we probably do not want to wast time trying to find its value.

  Let us assume that we already know:

  `int_1^infty1/x dx=infty`

  Let us look examine this uglier improper integral.

  `int_1^infty{4x^2+5x+8}/{3x^3-x-1} dx`

  By making the numerator smaller and the denominator bigger,

  `{4x^2+5x+8}/{3x^3-x-1} ge {3x^2}/{3x^3}=1/x`

  By Comparison Test, we may conclude that

  `int_1^infty{4x^2+5x+8}/{3x^3-x-1} dx` diverges.

  (Intuitively, if the smaller integral diverges, then the larger one has no chance to converge.)

  I hope that this was helpful.

  Wataru · · Oct 10 2014
• What is the Direct Comparison Test for Convergence of an Infinite Series?

  If you are trying determine the conergence of `sum{a_n}`, then you can compare with `sum b_n` whose convergence is known.

  If `0 leq a_n leq b_n` and `sum b_n` converges, then `sum a_n` also converges.
  If `a_n geq b_n geq 0` and `sum b_n` diverges, then `sum a_n` also diverges.

  This test is very intuitive since all it is saying is that if the larger series comverges, then the smaller series also converges, and if the smaller series diverges, then the larger series diverges.

  Wataru · · Aug 30 2014

• What is the Direct Comparison Test for Convergence of an Infinite Series?
• How do you use the direct comparison test for infinite series?
• How do you use the direct comparison test for improper integrals?
• How do you use the direct Comparison test on the infinite series `sum_(n=1)^oo5/(2n^2+4n+3)` ?
• How do you use the direct Comparison test on the infinite series `sum_(n=1)^ooln(n)/n` ?
• How do you use the direct Comparison test on the infinite series `sum_(n=2)^oon^3/(n^4-1)` ?
• How do you use the direct Comparison test on the infinite series `sum_(n=1)^oo9^n/(3+10^n)` ?
• How do you use the direct Comparison test on the infinite series `sum_(n=1)^oo(1+sin(n))/(5^n)` ?
• How do you use the direct Comparison test on the infinite series `sum_(n=1)^ooarctan(n)/(n^1.2)` ?
• How do you use basic comparison test to determine whether the given series converges or diverges for `sum n/sqrt(n^2-1)` from n=2 to `n=oo`?
• How do you use the Comparison Test to see if `1/(4n^2-1)` converges, n is going to infinity?
• How do you use the limit comparison test for `sum ((sqrt(n+1)) / (n^2 + 1))` as n goes to infinity?
• How do you test the series `(lnk)/(k^2)` for convergence?
• How do you use the comparison test to test for convergence if `1 / (xcosx) dx` from 0 to `pi/2`?
• How would I use the Comparison Test in calculus to solve the integral `(cos(4x) +3) / (3x^3 + 1)` from 1 to infinity?
• How do you use the comparison test for `sum 1/(5n^2+5)` for n=1 to `n=oo`?
• How do you use the convergence tests, determine whether the given series converges `sum (7-sin(n^2))/n^2+1` from n to infinity?
• How do you know if `sumn/(e(n^2))` converges from 1 to infinity?
• How do you test for convergence for `sum (3n+7)/(2n^2-n)` for n is 1 to infinity?
• How do you test for convergence: `int (((sin^2)x)/(1+x^2)) dx`?
• How do you test for convergence for `sum((-1)^n)*(sqrt(n))*(sin(1/n))` for n is 1 to infinity?
• How do you test for convergence for `sum(4 + abs(cosk))/(k^3)` for k=1 to infinity?
• How do you test for convergence for `sum sin(n*(pi)/2) / (n!)` for n=1 to infinity?
• How do you test the convergence of the series `cos(n) sin (pi/n)^2`?
• How do you show the convergence of the series `(n!)/(n^n)` from n=1 to infinity??
• How do you test for convergence for `(-1)^(n-1) /(2n + 1)` for n=1 to infinity?
• How do you test for convergence for `(sin 4n)/(4^n)` for n=1 to infinity?
• How do you test for convergence for `int (lnx)/x^2dx` from 1 to infinity?
• How do you test for convergence for `sum ln(n)/n^2` for n=1 to infinity?
• How do you test for convergence `(sin(2n))/(1+(2^n))` from n=1 to infinity?
• How do you test the series `(n / (5^n) )` from n = 1 to infinity for convergence?
• How do you test for convergence for `sum(5^k+k)/(k!+3)` from k=1 to infinity?
• How do you test the series for convergence or divergence for `sum (-1)^n n / 10^n` from n=1 to infinity?
• How do you determine if series `1/n!` converge or diverge with comparison test?
• How do you know if the series `sum 1/(n^(1+1/n))` converges or diverges for (n=1 , ∞) ?
• How do you know if the series `(1+n+n^2)/(sqrt(1+(n^2)+n^6))` converges or diverges for (n=1 , ∞) ?
• How do you know if the series `((-1)^n)*sqrt(n)/(ln(n))` converges or diverges for (n=2, ∞) ?
• How do you know if the series `(1/(2n+1))` converges or diverges for (n=1, ∞) ?
• How do you use the direct comparison test to determine if `Sigma 1/(n^2+1)` from `[1,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma 1/(3n^2+2)` from `[1,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma 1/(n-1)` from `[1,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma 1/(sqrtn-1)` from `[2,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma 1/(3^n+1)` from `[0,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma 3^n/(4^n+5)` from `[0,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma lnn/(n+1)` from `[2,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma 1/(sqrt(n^3+1))` from `[1,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `Sigma 1/(n!)` from `[0,oo)` is convergent or divergent?
• How do you use the direct comparison test to determine if `sume^(-n^2)` from `[0,oo)` is convergent or divergent?
• Find the limit of the sequence `ln(n^2+2) -1/2ln(n^4+4)`. Does it converge to 0?
• Determine whether the series `sum_(n=1)^oo (2n^2 +3n)/sqrt(5+n^5)` is convergent or divergent. How do i tell which comparison test to use?
• Does the series converge or diverge?