Expert: Paul Klarreich - 6/23/2010

Question
For what values of p does summation  n/sqrt(4+n^p) converge? (Indicate reasoning.

Answer
Questioner: Krystal
Country: United States
Category: Calculus
Private: No
Subject: Help ASAP Please
Question: For what values of p does summation  n/sqrt(4+n^p) converge? (Indicate reasoning.
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This appears to be asymptotic: (I will use r instead of p, for now)
   n
----------- =
sqrt(4+n^r)

   1
-------------------- ==> (for large n)
sqrt(4/n^2 + n^r-2)

   1
----------------- ==> (for large n)
sqrt(0 + n^r-2)

   1
--------------- ==> (for large n)
sqrt(n^r-2)

   1
------------ ==> (for large n)
n^(r/2 - 1)

Now this is the so-called 'p-series'.  It converges for p > 1, so:

r/2 - 1 > 1 is required, which you can finish up.