The integral test basically works from the definition of the integral (quick version: the integral is the accumulated sum of infinitely thin differential intervals ##dn## over a specified interval ##a->b##).
A paraphrased version of the integral test is as follows:
Let there be a function ##f(n) = a_n## where ##a_n## is a series lying within the domain ##[k,oo)##. There exists another function ##f(x)## that is continuous, positive, and decreasing such that the convergence or divergence of ##int_k^(oo)f(x)dx## determines the convergence or divergence of ##sum_(n=k)^(oo)a_n##, respectively.
So, essentially, we have to integrate this, which is indeed continuous, positive, and decreasing at ##[1,oo)##:
##int_1^(oo)1/(sqrt(x+1))dx##
We can do that like so:
##= int_1^(oo)(x+1)^(-“1/2”)dx##
##= |[2(x+1)^(“1/2”)]|_1^(oo)##
At this point we know that ##sqrt(x+1)## is a constantly increasing function, so it has an “open” accumulation that can never stop without a well-defined right-end boundary. Basically, it’s a half-open integral that extends its domain forever and so it has no finite area.
##= 2(oo)^(“1/2”) – cancel(2(1+1)^(“1/2”))^”small”##
##=> oo##
The integral does not converge, and so the series does not converge either. QED
Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of HIGH QUALITY & PLAGIARISM FREE. To make an Order you only need to click Ask A Question and we will direct you to our Order Page at WriteDemy. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Fill in all the assignment paper details that are required in the order form with the standard information being the page count, deadline, academic level and type of paper. It is advisable to have this information at hand so that you can quickly fill in the necessary information needed in the form for the essay writer to be immediately assigned to your writing project. Make payment for the custom essay order to enable us to assign a suitable writer to your order. Payments are made through Paypal on a secured billing page. Finally, sit back and relax.
About Writedemy
We are a professional paper writing website. If you have searched a question and bumped into our website just know you are in the right place to get help in your coursework. We offer HIGH QUALITY & PLAGIARISM FREE Papers.
How It Works
To make an Order you only need to click on “Place Order” and we will direct you to our Order Page. Fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Are there Discounts?
All new clients are eligible for 20% off in their first Order. Our payment method is safe and secure.