clairez93 said:Consider:
[tex]\int^{\infty}_{1}\;\frac{x^{k-1}}{x^{k}+c} dx[/tex]
Not sure how to integrate this expression to determine if it converges or diverges.
The Integral Test is a method used to determine whether an infinite series converges or diverges. It states that if a series has a corresponding continuous, positive, and decreasing function, then the series and the integral of that function will either both converge or both diverge.
To use the Integral Test, you first need to find a continuous, positive, and decreasing function that represents the series. Then, you take the integral of that function and evaluate it from 1 to infinity. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.
No, the Integral Test can only be used for series with positive terms. If a series has negative terms or alternating signs, the Integral Test cannot be applied. In those cases, other tests such as the Alternating Series Test or Ratio Test can be used.
The Integral Test is a more general version of the p-Series Test. The p-Series Test is a special case of the Integral Test, where the corresponding function is f(x) = 1/x^p. So, if the integral of 1/x^p converges, then the series 1/n^p also converges, and vice versa.
No, the Integral Test only tells you whether a series converges or diverges, but it does not give the actual sum of the series. To find the sum of a convergent series, you can use other methods such as the Geometric Series formula or the Summation formula.