Root Test and Integral Test Question

[RJLiberator]

Homework Statement

[/B]
From K=4 to infinity the Σ (-1)^k (k/e^k)

Converge or diverge?
Use:
a) Ratio Test
b) Root Test
c) Integral Test
d) Alternating series test

Homework Equations

The Attempt at a Solution

For the alternating series test and ratio test I have the correct answer that it converges. These were fairly simple for me to proceed with.
However, I am stuck on the Root test and Integral test.
For the root test I DID get an answer, but it seems corrupt:

Lim as n approaches infinity of (|(-1)^k (k/e^k|))^(1/k)
With some simplification I narrowed it down to
The lim as n-->infinity of (|n^(1/n)|/e)

Which doesn't seem solvable ?

And for the Integral test, I am seeing the answer requires imaginary numbers, etc. which we do not use in this class. Is it possible that the instructor did not realize this? Does this problem demand the use of imaginary numbers, etc? If so, I imagine I would be able to pass on this part.

Thanks for any guidance.

 

[mfb]

Root test: it is solvable. The numerator is a well-known limit problem with a standard answer, but here you do not need the exact limit - it is sufficient to find some upper bound, and that is easier to find.

And for the Integral test, I am seeing the answer requires imaginary numbers, etc.

You can show absolute convergence instead of the (weaker) convergence. There, you don't get issues with complex numbers.

 

[RJLiberator]

Root test: it is solvable. The numerator is a well-known limit problem with a standard answer, but here you do not need the exact limit - it is sufficient to find some upper bound, and that is easier to find.

You can show absolute convergence instead of the (weaker) convergence. There, you don't get issues with complex numbers.

Ah, music to my ears. I see exactly what to do with the absolute convergence of the integral test. I took out the (-1)^k and then integrated to get a result o 5/e^4 which concludes absolute convergence.

Now, I will try to work on the root test.

Thank you for your guidance.

 

[RJLiberator]

Root test: it is solvable. The numerator is a well-known limit problem with a standard answer, but here you do not need the exact limit - it is sufficient to find some upper bound, and that is easier to find.

You can show absolute convergence instead of the (weaker) convergence. There, you don't get issues with complex numbers.

For the Root Test:

I took the limit of the numerator and denominator. For the common limit of n^(1/n) the limit is 1. For e, the limit is the constant --> e.
Thus, answer is 1/e and the limit is less then 1 meaning absolute convergence.

Thank you for your guidance.