Strategies to Test an Infinite Series for Convergence

Key Questions

• If I want to test the series `sum_(n=1)^oo(n^2+2^n)/(1-e^(n+1))` for convergence, what would be the best test to use and why?

  By pulling out the negative sign so that `a_n ge0`, let

  `-sum_{n=1}^inftya_n=-sum_{n=1}^infty{n^2+2^n}/{e^{n+1}-1}`.

  I would use Limit Comparison Test since we can make a ball-park estimate of the series by only looking at the dominant terms on the numerator and the denominator. This series can be compared to

  `sum_{n=1}^inftyb_n=sum_{n=1}^infty{2^n}/{e^{n+1}}=sum_{n=1}^infty1/e(2/e)^n`,

  which is a convergent geometric series with `|r|=|2/e|<1`.

  Let us make sure that they are comparable.

  `lim_{n to infty}{a_n}/{b_n}=lim_{n to infty}{{n^2+2^n}/{e^{n+1}-1}}/{{2^n}/{e^{n+1}}}`

  `=lim_{n to infty}{{n^2}/{2^n}+1}/{1-1/e^{n+1}}={0+1}/{1-0}=1 < infty`

  (Note: `lim_{n to infty}{n^2}/2^n=0` by applying l'Hopital's Rule twice.)

  So, `sum_{n=1}^infty a_n` converges by Limit Comparison Test.

  Hence, `sum_{n=1}^infty{n^2+2^n}/{1-e^{n+1}}` also converges since negation does not affect the convergence of the series.

  ---

  I hope that this was helpful.

  Wataru · · Oct 19 2014
• When testing for convergence, how do you determine which test to use?

  There is no general method of determining the test you should use to check the convergence of a series.

  • For series where the general term has exponents of `n`, it's useful to use the root test (also known as Cauchy's test).
    Example 1: Power Series
    The definition of the convergence radius of the of a power series comes from the Cauchy test (however, the actual computation is usually done with the following test).

  • Generally, the computation of the ratio test (also known as d'Alebert's test) is easier than the computation of the root test.
    Example 2: Inverse Factorial
    For the series `sum_(n=1)^(oo) 1/(n!)` the d'Alembert's test gives us:
    `lim_(n to oo) |1/((n+1)!)|/|1/(n!)| = lim_(n to oo) |n!|/(|(n+1)!|) = lim_(n to oo) |(n!)/((n+1)n!)| = lim_(n to oo) |1/(n+1)| = 0`
    So the series is convergent.

  • If you know the result of the improper integral of the function `f(x)` such that `f(n)=a_n`, where `a_n` is the general term of the series being analyzed, then it might be a good idea to use the integral test.
    Example 3: A proof for the Harmonic Series.
    Knowing that the improper integral `int_1^(oo) 1/x dx` is divergent (it's easy to check) implies that the harmonic series `sum_(n=1)^(oo) 1/(n)` diverges.

  • Comparision tests are only useful if you know an appropriate series to compare the one you're analyzing to. However, they can be very powerful.
    Example 4: Hyperharmonic Series
    The series of the form `sum_(n=1)^(oo) 1/(n^p)` are called hyperharmonic series or `p`-series. If you can show that the series `sum_(n=1)^(oo) 1/(n^(1+epsilon))` converges, for some small, positive value of `epsilon`, than any `p`-series such that `p>1 + epsilon` converges.

  Vinicius M. G. Silveira · 2 · Dec 22 2014

• What is the sum of the series `1+ln2+(((ln2)^2)/(2!))+...+(((ln2)^n)/(n!))+...`?
• How do I write `(5/(1*2))+(5/(2*3))+(5/(3*4))+...+(5/n(n+1))+...`in summation notation, and how can I tell if the series converges?
• How do you show whether the improper integral `int ln(x)/x^3 dx` converges or diverges from 1 to infinity?
• How do you show whether the improper integral `int e^x/ (e^2x+3)dx` converges or diverges from 0 to infinity?
• How do you show whether the improper integral `int (79 x^2/(9 + x^6)) dx` converges or diverges from negative infinity to infinity?
• How do you show whether the improper integral `int 1/ (1+x^2) dx` converges or diverges from negative infinity to infinity?
• How do you prove that the integral of ln(sin(x)) on the interval [0, pi/2] is convergent?
• How do you show whether the improper integral `int (x^2)(e^(-x^3)) dx` converges or diverges from negative infinity to infinity?
• Using the definition of convergence, how do you prove that the sequence `{5+(1/n)}` converges from n=1 to infinity?
• Using the definition of convergence, how do you prove that the sequence `{2^ -n}` converges from n=1 to infinity?
• Using the definition of convergence, how do you prove that the sequence `lim 1/(6n^2+1)=0` converges?
• Using the definition of convergence, how do you prove that the sequence `lim (3n+1)/(2n+5)=3/2` converges?
• Using the definition of convergence, how do you prove that the sequence `lim 2/(sqrt(n+3))=0` converges?
• Using the definition of convergence, how do you prove that the sequence `(-1)^n/(n^3-ln(n))` converges from n=1 to infinity?
• Using the definition of convergence, how do you prove that the sequence `limit (sin n)/ (n) = 0` converges from n=1 to infinity?
• Using the definition of convergence, how do you prove that the sequence `lim (n + 2)/ (n^2 - 3) = 0` converges from n=1 to infinity?
• Question #9c196
• Question #a65c6
• How do you test the series `Sigma n/(n+4)` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma 2/(4n-3)` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma (n+1)/n^3` from n is `[1,oo)` for convergence?
• How do you test the series `sum_(n=1)^oo n/(n^2+2)` for convergence?
• How do you test the series `Sigma 1/((n+1)(n+2))` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma 1/sqrt(n(n+1))` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma (n+3)/(n(n+1)(n-2))` from n is `[3,oo)` for convergence?
• How do you test the series `Sigma n/((n+1)(n^2+1))` from n is `[0,oo)` for convergence?
• How do you test the series `sum_(n=0)^(oo) n/((n+1)(n+2))` for convergence?
• How do you test the series `Sigma 1/(nsqrtn)` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma sqrt(n+1)-sqrtn` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma (sqrt(n+2)-sqrtn)/n` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma (3n^2+1)/(2n^4-1)` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma 1/sqrt(n^3+4)` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma n/sqrt(n^3+1)` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma sqrtn/(3n+2)` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma n^-n` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma 1/(2^n-n)` from n is `[1,oo)` for convergence?
• How do you test the series `sum_(n=1)^(oo) sin^2n/n^2` for convergence?
• How do you test the series `Sigma n^2/2^n` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma rootn(n)/n` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma rootn(n)/n^2` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma 1/(n!)` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma 5^n/(3^n+4^n)` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma (5^n+6^n)/(2^n+7^n)` from n is `[0,oo)` for convergence?
• How do you test the series `Sigma 1/(2+lnn)` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma lnn/n` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma lnn/(nsqrtn)` from n is `[1,oo)` for convergence?
• How do you test the series `Sigma 1/(nlnn)` from n is `[2,oo)` for convergence?
• How do you test the series `Sigma 1/(ln(n!))` from n is `[2,oo)` for convergence?
• How do you determine if `a_n = (1+1/n^2)^n` converge and find the limits when they exist?
• How do you determine if `a_n(1+1/sqrtn)^n` converge and find the limits when they exist?
• How do you determine if `a_n=(1+n)^(1/n)` converge and find the limits when they exist?
• How do you test for convergence of `Sigma (3n-7)/(10n+9)` from `n=[0,oo)`?
• How do you test for convergence of `Sigma 5/(6n^2+n-1)` from `n=[1,oo)`?
• How do you test for convergence of `Sigma n e^-n` from `n=[1,oo)`?
• How do you test for convergence of `Sigma (ln(n))^-n` from `n=[2,oo)`?
• How do you test for convergence of `sum_(n=2)^(oo) lnn^(-lnn)`?
• How do you test for convergence of `Sigma (-1)^n/sqrt(lnn)` from `n=[3,oo)`?
• How do you test for convergence of `Sigma (-1)^n(1-n^2)` from `n=[1,oo)`?
• How do you test for convergence of `Sigma (-1)^n n^(-1/n)` from `n=[1,oo)`?
• How do you test for convergence given `Sigma (-1)^n(1-1/n^2)` from `n=[1,oo)`?
• How do you test for convergence given `Sigma (-1)^n n^(-1/n)` from `n=[1,oo)`?
• How do you find the radius of convergence of the power series `Sigma 2^n n^3 x^n` from `n=[0,oo)`?
• How do you find the radius of convergence of the power series `Sigma x^n/(n!)^(1/n)` from `n=[1,oo)`?
• How do you find the radius of convergence of the power series `Sigma (n!)/(n^n)x^(2n)` from `n=[1,oo)`?
• How do you find the interval of convergence of `Sigma (x+10)^n/(lnn)` from `n=[2,oo)`?
• Question #41dc8
• How do you test the improper integral `int x^-2 dx` from `[2,oo)` and evaluate if possible?
• How do you test the improper integral `int x^-0.9 dx` from `[0,1]` and evaluate if possible?
• How do you test the improper integral `int x^(-1/2) dx` from `[1,oo)` and evaluate if possible?
• How do you test the improper integral `int (2x-1)^-3dx` from `(-oo,0]` and evaluate if possible?
• How do you test the improper integral `int (2x-1)^-3dx` from `[0,1/2]` and evaluate if possible?
• How do you test the improper integral `int x^(-1/3)dx` from `[-1,0]` and evaluate if possible?
• How do you test the improper integral `int (x^2+2x-1)dx` from `[0,oo)` and evaluate if possible?
• How do you test the improper integral `int (x^-2-x^-3)dx` from `[3,oo)` and evaluate if possible?
• How do you test the improper integral `int (x(1+x^2)^-2)dx` from `[0,oo)` and evaluate if possible?
• How do you test the improper integral `int(x-1)^(-2/3)dx` from `[0,1]` and evaluate if possible?
• How do you test the improper integral `int x^-2dx` from `[-1,1]` and evaluate if possible?
• How do you test the improper integral `int x^(-2/3)dx` from `[-1,1]` and evaluate if possible?
• How do you test the improper integral `int x/sqrt(1-x^2)dx` from `[0,1]` and evaluate if possible?
• How do you test the improper integral `int 2x(x^2-1)^(-1/3)dx` from `[0,1]` and evaluate if possible?
• How do you test the improper integral `int 2x^-3dx` from `[-1,1]` and evaluate if possible?
• How do you test the improper integral `int (2x-1)^(-2/3)dx` from `[0,1]` and evaluate if possible?
• How do you test the improper integral `int (3x-1)^-5dx` from `[0,1]` and evaluate if possible?
• How do you test the improper integral `intx^2 dx` from `(-oo, oo)` and evaluate if possible?
• How do you test the improper integral `int (2x-1)^3 dx` from `(-oo, oo)` and evaluate if possible?
• How do you test the improper integral `int x^(-1/3) dx` from `(-oo, oo)` and evaluate if possible?
• How do you test the improper integral `int x^3 dx` from `(-oo, oo)` and evaluate if possible?
• How do you test the improper integral `int x^(-3/2) dx` from `[0, oo)` and evaluate if possible?
• How do you test the improper integral `int (3x)/(x+1)^4 dx` from `[0, oo)` and evaluate if possible?
• How do you test the improper integral `int absx(x^2+1)^-3 dx` from `(-oo, oo)` and evaluate if possible?
• How do you test the improper integral `int (2x)/(sqrt(x^2+1)) dx` from `(-oo, oo)` and evaluate if possible?
• How do you test the improper integral `int (x-1)^-2+(x-3)^-2 dx` from `[1,3]` and evaluate if possible?
• How do you test the improper integral `int x/absxdx` from `[-5,3]` and evaluate if possible?
• How do you test the improper integral `int sintheta/cos^2theta` from `[0,pi/2]` and evaluate if possible?
• How do you test the improper integral `int sintheta/sqrtcostheta` from `[0,pi/2]` and evaluate if possible?
• `sum_(n=0)^oo 5^n/(3^n +4^n)`. Does the series converge or diverge?
• Is it possible to for an integral in the form `int_a^oo f(x)\ dx`, and `lim_(x->oo)f(x)!=0`, to still be convergent?
• Find the values of `x` for which the following series is convergent?
• Is the series `\sum_(n=1)^\inftyn^2/(n^3+1)` absolutely convergent, conditionally convergent or divergent?
• Is the series `\sum_(n=1)^\infty\tan^-1(1/n)` absolutely convergent, conditionally convergent or divergent?
• Is the series `\sum_(n=1)^\infty((-5)^(2n))/(n^2 9^n)` absolutely convergent, conditionally convergent or divergent?
• Is the series `\sum_(n=1)^\infty n e^(-n)` absolutely convergent, conditionally convergent or divergent?
• Is the series indicated absolutely convergent, conditionally convergent, or divergent? `rarr\4-1+1/4-1/16+1/64...`
• Is the series `\sum_(n=0)^\infty1/((2n+1)!)` absolutely convergent, conditionally convergent or divergent?