You are Here: Home >< Maths

Convergence watch

1. I have to find if the following series converge:

Sum from k=1 to k=inf of (1/sqrt(k))*e^(-2*sqrt(k))

I tried to do it with the ratio test and found that abs(a_(k+1)/a_k) = abs(sqrt(k/(k+1)) * e^ (2*(sqrt(k) * sqrt(k+1))
which tends to 1 as k tends to infinity, so I can't deduce if the sequence converges or diverges using the ratio test.

Do you recommend any other way? I don't know with what test to handle it, thanks in advance!
2. Cauchy condensation test should work I think.
3. We haven't learned this test in our module. Can you give me some more information? I know about comparison test , ratio test (which is not working in this case) and the alternating series test. I don't know what to compare using the comparison test or if the alternating series test is suitable. Thanks in advance for any help!
4. Integral test, maybe? Havn't got any paper handy, but..

Function is positive, continuous and decreasing, so it's viable. Maybe.

_Kar.
5. The comparison test is always "suitable"; however finding a suitable series to compare against (that you can show converges/diverges) may be very hard.

Ratio/root/alternating series tests definitely don't work.

It's easy enough to google the Cauchy test.
6. Unless I'm missing something a comparison test will kill it very quickly : I'm assuming the question is in which case you can focus on the thing that goes smallest (the e) and ignore the 1\sqrt(k)
7. You still need to show converges. I felt the quickest way was condensation, but if you have another approach that's fine.
8. So I will Google about condensation test and try to put it in practice, thank you very very much. If I still have any problems I will write you again! Thank you everyone!
9. (Original post by DFranklin)
You still need to show converges. I felt the quickest way was condensation, but if you have another approach that's fine.
Spoiler:
Show
is quick if you permit various properties of exp

I would have said that doing this is better than using a test the op hasn't learnt but you might disagree and say that the properties I am assuming are too strong
10. (Original post by IrrationalNumber)
Spoiler:
Show
is quick if you permit various properties of exp

I would have said that doing this is better than using a test the op hasn't learnt but you might disagree and say that the properties I am assuming are too strong
A bit 6 of one and half-dozen of the other I guess. In the order I learned analysis you wouldn't have proved this about exp until well after doing all the convergence tests, but that certainly doesn't make what you've done invalid.

Although I will admit to a sneaking feeling that everyone should learn about the Cauchy condensation test.
11. (Original post by Darkprince)
I have to find if the following series converge:

Sum from k=1 to k=inf of (1/sqrt(k))*e^(-2*sqrt(k))

I tried to do it with the ratio test and found that abs(a_(k+1)/a_k) = abs(sqrt(k/(k+1)) * e^ (2*(sqrt(k) * sqrt(k+1))
which tends to 1 as k tends to infinity, so I can't deduce if the sequence converges or diverges using the ratio test.

Do you recommend any other way? I don't know with what test to handle it, thanks in advance!
USe the integral test
This series is monotone decreasing and defined on the
open interval so it converges iif
is finite.
This itegral is a simple case if you consider that with a minus sign it will be the
integral, of
which antiderivative is
12. In hindsight, the factor of 2 in the exponent looks like it was put there to make the function easy to integrate, so I'd be very surprised if ztibor's method wasn't the one expected.
13. (Original post by DFranklin)
In hindsight, the factor of 2 in the exponent looks like it was put there to make the function easy to integrate, so I'd be very surprised if ztibor's method wasn't the one expected.
It is not the 2 but the factor make easy to integrate
because the difference between it and the derivative of
in the exponent is only a constant factor
14. (Original post by ztibor)
It is not the 2 but the factor make easy to integrate
because the difference between it and the derivative of
in the exponent is only a constant factor
Well yes, it's impossible to integrate without the 1/sqrt(x) term.

But it's the 2 in the exponent that "doesn't belong there". If you were just looking at convergence, there's "no point" in multiplying by 2. So it looks like it's only there to make it come out nicely when you integrate, and it's a bit of (circumstantial) evidence that integration was the intended method.
15. I did it using the comparison method, thanks for your replies

Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: March 25, 2011
Today on TSR

Exam Jam 2018

Join thousands of students this half term

Poll
Useful resources

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

How to use LaTex

Writing equations the easy way

Study habits of A* students

Top tips from students who have already aced their exams

Chat with other maths applicants