Integral Test for Convergence of an Infinite Series
The Integral Test is a powerful tool in the realm of calculus used to determine the convergence or divergence of an infinite series by relating it to an improper integral. By establishing a connection between the series and the integral of a continuous function, the Integral Test provides a straightforward method for assessing convergence. This method is particularly useful when dealing with series whose terms are positive, decreasing, and continuous functions of a variable. Through analysis of the integral, mathematicians can effectively ascertain the behavior of the corresponding series, facilitating deeper understanding and application within various mathematical contexts.
Questions
- How do you use the integral test to determine whether #int e^(-x^2)# converges or diverges from #[0,oo)#?
- Using the integral test, how do you show whether #sum1/[(n^2)+4)# diverges or converges from n=1 to infinity?
- Why does the integral test not apply to #Sigma (-1)^n/n# from #[1,oo)#?
- How do you find the positive values of p for which #Sigma lnn/n^p# from #[2,oo)# converges?
- 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?
- What is the Integral Test for Convergence of an Infinite Series?
- 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^3# diverges or converges from n=1 to infinity?
- How do you use the integral test to determine the convergence or divergence of #1+1/4+1/9+1/16+1/25+...#?
- Using the integral test, how do you show whether #sum 1 / [sqrt(n) * (sqrt(n) + 1)]# diverges or converges from n=1 to infinity?
- How do you determine convergence or divergence for the summation of #n*e^(-n/2)# using the integral test how do you answer?
- How do you use the integral test to determine whether #int x^-x# converges or diverges from #[1,oo)#?
- Evaluate the integral or show that it is divergent?
- How do you determine if the improper integral converges or diverges #int 5x^(2)e^(-x^(3))# from 1 to infinity?
- Using the integral test, how do you show whether #n/(n^2+1)# diverges or converges?
- 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 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?
- How do you find the positive values of p for which #Sigma n/(1+n^2)^p# from #[2,oo)# converges?
- How do you determine if the improper integral converges or diverges #int ln(sin(x))# from 0 to pi/2?
- Using the integral test, how do you show whether #sum (1/n^2)cos(1/n) # diverges or converges from n=1 to infinity?