Do You Adjust Bounds When Using the Integral Test for Series Convergence?

[RJLiberator]

Homework Statement

When using the Integral Test do you need to change the bounds to n+1 and n-1 for an increasing and decreasing function respectively?

This is a question that comes up when using the integral test.

I think that you just use the original bounds for the integral. We are a bit confused with something that the teacher wrote and I just wanted clarifying.

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

When using the Integral Test do you need to change the bounds to n+1 and n-1 for an increasing and decreasing function respectively?

This is a question that comes up when using the integral test.

I think that you just use the original bounds for the integral. We are a bit confused with something that the teacher wrote and I just wanted clarifying.

Homework Equations

The Attempt at a Solution

I don't understand your question. The integral test is used to determine the convergence (or divergence) of a series of nonnegative terms. Part of the "fine print" for this test is that the function must be monotone decreasing.

I think that you just use the original bounds for the integral.

What original bounds? You're starting from an infinite series.

 

[RJLiberator]

Thank you for your reply.

Let me make this more clear.

Wiki article: https://en.wikipedia.org/wiki/Integral_test_for_convergence

Under the 'proof' section of this article it changes the bounds to n+1 and n-1 under certain occasions.
During our lecture, our instructor introduced us to this test this way.
A fellow student and I were conversing over how to use the integral test properly.
He indicated that since the function is decreasing you need to subtract 1 from the lower bound according to our notes based off this.
However, on every application/online site I do not see people subtracting or adding one to the bounds.

For example:

If you were taking the series of 1/k^2 from k = 5 to k = infinity, and wanted to do the integral test, would you set the integral up from 5 to infinity or from 4 to infinity?

 

[Mark44]

Thank you for your reply.

Let me make this more clear.

Wiki article: https://en.wikipedia.org/wiki/Integral_test_for_convergence

Under the 'proof' section of this article it changes the bounds to n+1 and n-1 under certain occasions.
During our lecture, our instructor introduced us to this test this way.
A fellow student and I were conversing over how to use the integral test properly.
He indicated that since the function is decreasing you need to subtract 1 from the lower bound according to our notes based off this.
However, on every application/online site I do not see people subtracting or adding one to the bounds.

For example:

If you were taking the series of 1/k^2 from k = 5 to k = infinity, and wanted to do the integral test, would you set the integral up from 5 to infinity or from 4 to infinity?

It doesn't really make any difference whether you integrate from 4 to ∞ or from 5 to ∞. If the definite integral comes out to a number, your series converges, and if the integral is unbounded, then the series diverges.

 

[mfb]

For the integral test, you are checking the convergence of the integral. Any finite contribution from small numbers is not relevant.

For the proof, you need some index manipulation, but this is not relevant for the application any more. If it helps, you can even start your integration at x=10000.

Edit: Mark was faster by a few seconds.

 

[RJLiberator]

Thanks guys for the replies.

So, I can conclude then that in our particular situation, it was not necessarily right to subtract 1 from the lower boundary, although it does not make the integral test wrong.

Thank you.