Convergence Tests: 3 Problems Explained

[rjs123]

Homework Statement

Determine if the following converge:

a. (∞,n=1) ∑ (1+1/n)^n

b. (∞,n=1) ∑ sin(n)/(n^2 + √n)

c. (∞,n=3) ∑ 1/(k ln^2 k)

The Attempt at a Solution

a. I tried the root test, but it failed, so i immediately went to the limit divergence test...ended up getting 1^inf...not sure what to do next...possibly p-test?

b. Ended up doing a direct comparison test...convergence...p>1

c. tried limit divergence test...got 0...stuck on what other test to do...maybe ratio test?

 

[SVXX]

a. Use Cauchy's Fundamental test for divergence. The limit when n tends to infinity for this one is 'e'. Check the conditions for the test I mentioned and you'll get your answer. (The test being http://www.math.oregonstate.edu/hom...StudyGuides/SandS/SeriesTests/divergence.html)

b. Seems fine to me if you used the comparison test, I had the same thing in mind.

c. Use Cauchy's Integral test.

 

[rjs123]

Ok, I've completed a and b but I am stuck on the integration part of c, any help would be appreciated.

 

[SVXX]

Employ this nifty little substitution : take ln(k) = t and so dk/k = dt...

 

[rjs123]

Employ this nifty little substitution : take ln(k) = t and so dk/k = dt...

I was close...i set u=ln^2 k...instead of just ln(k)...ended up getting infinity...so the series diverges based on the integral test.

 

[SVXX]

You made a tiny mistake in putting the limits into the integrated expression...the integral is infact finite and the series converges :)

 

[rjs123]

You made a tiny mistake in putting the limits into the integrated expression...the integral is infact finite and the series converges :)

I'm a little confused...I integrated this...

Int(3, inf) 1/(kLn^2 k) dk


ended up with [1/3u^3](inf, 3)

= inf - a #...which is inf

 

[SVXX]

Hmm, no. If you take only ln(k) = t and differentiate to get dk/k = dt, that means 1/k * 1/(lnk)^2 dk = dt/t^2. The limits change to log3 to infinity. Integration of 1/t^2 is -1/t. Can you finish it off?

 

[rjs123]

Hmm, no. If you take only ln(k) = t and differentiate to get dk/k = dt, that means 1/k * 1/(lnk)^2 dk = dt/t^2. The limits change to log3 to infinity. Integration of 1/t^2 is -1/t. Can you finish it off?

Ok...i see where i went wrong...thanks.