11.3) Determine whether the series is convergent or divergent. Please show a step by step solution.

Question

Determine whether the series is convergent or divergent. Please show a step by step solution.

Answer 1

Hi,

for saying converge or diverge of this ∑ ln n / n^p ,

You need to know when

a) p>1

b) positive terms

c) continuous and f'(x) < 0

always ∑ ln n / n^p , is converge your question is the same and answer is CONVERGE because p = 2 and try to take derivative that you will see it is negative for x>1 that hear you have n>2.

Note: if p ≤ 0 lim ln n/ n^p = ∞ ( when n→∞) and the series DIVERGE

I hope it was useful,

Minoo

Answer 2

ln2/2^2 + ln3/3^2 + ln4/4^2 + ln5/5^2 + ln6/6^2+ ln7/7^2 + ln8/8^2+ ln9/9^2 + ln10/10^2+ ln11/11^2 +ln12/12^2 + ...

.693/4 + 1.099/9 + 1.39/16 +1.609/25+ 1.79/36 + 1.95/49 + 2.08/64 + 2.2/81+ 2.3/100 + 2.4/121 +2.5/144 ...'

= .173 + .122 + .087. + .064+ .05 + .04 + .03+ .03+ .02 + .02 + .02 .. (seems to be getting ever smaller, term by term)

= .656 + ... = likely less than one and likely converges, but might diverge

use integral test, find the integral of ln(n)/n^2 dn

requires integration by parts

-lnx/x -1/x evaluated from 2 to infinity =a constant = about 1, converges