Hey guys, can someone help me with the following proofs:

 

Prove that the sum (1/n^p) for n=1 to infinity converges for p>1.

 

Also, can you help me in proving the ratio test, that is suppose that x_n > 0 for all n in N (N = set of natural numbers) and suppose also that lim sup ( x_n+1/x_n) < 1, then prove that the series x_n for n = 1 to infinity converges. Then if lim inf ( x_n+1/x_n) > 1 then the same series x_n diverges.

Integral test.

 

In the second there are I'm sure many explanations of this, but it simply boils down to replacing the orginal with a Geometric Series. You ought to try using that hint to prove it yourself.

For the first one I cannot use the integral test because we haven't proven it yet. I believe that I need to show that the sequence of partial sums is bounded to show that the series converges, I tried hard but I couldn't finish my proof.

You could show that n^r < r^n for a fixed r>1 and all n sufficiently large. Of course if you don't tell us what techniques you are allowed to adopt then we're going to struggle to find you any proof that is acceptable.

This is a very famous problem that can be solved by integral test that is shown in the following link:

 

http://www.scienceoxygen.com/mathnote/calculus221.html

 

But you are not allowed to use "integral test", you might use "Cauchy criterion" to prove it if it is allowed. You can use the technique shown in the following link:

 

http://www.scienceoxygen.com/mathnote/seq202.html