Integral Test: Analyzing 1/(n+1)^x with x > 1

In summary, the conversation is about using the integral test on a series with x>1 and the confusion about evaluating the integral due to the presence of x. The solution is suggested to substitute x with a and write down the attempt.
  • #1
brunette15
58
0
Hi,
I am trying to use an integral test on the following series:

The sum from n=0 to infinity on 1/(n+1)^x where x>1

I know the process of using the integral test however i am unsure as to how to evaluate the integral with the x in the series :/

Thanks in advance!
 
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  • #2
Maybe using $x$ confuses you. Try putting $x=a$, what do you have? Can you write down your attempt?
 
  • #3
ZaidAlyafey said:
Maybe using $x$ confuses you. Try putting $x=a$, what do you have? Can you write down your attempt?

Yes i think using x was a little confusing. I was able to get it thankyou!
 

FAQ: Integral Test: Analyzing 1/(n+1)^x with x > 1

What is the integral test?

The integral test is a method used in calculus to determine whether an infinite series converges or diverges. It involves comparing the given series to a related improper integral and using properties of integrals to determine the convergence or divergence of the series.

How is the integral test used to analyze 1/(n+1)^x with x > 1?

The integral test for 1/(n+1)^x with x > 1 involves finding the improper integral of the series using the power rule for integrals. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

What is the power rule for integrals?

The power rule for integrals states that the integral of x^n is equal to (x^(n+1))/(n+1) + C, where C is a constant. This rule is used to evaluate improper integrals of the form 1/(n+1)^x with x > 1.

Can the integral test be used for series with negative terms?

Yes, the integral test can be used for series with negative terms. When evaluating the improper integral, the absolute value of the terms should be used to determine the convergence or divergence of the series.

Are there any other tests to determine the convergence or divergence of a series?

Yes, there are other tests such as the comparison test, ratio test, and root test that can be used to determine the convergence or divergence of a series. These tests may be more suitable for certain types of series, so it is important to consider which test is most appropriate for a given series.

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