# Calculus

posted by .

For what values of p>0 does the series

Riemann Sum [n=1 to infinity] 1/ [n(ln n) (ln(ln n))^p]

converge and for what values does it diverge?

You need to let the summation start at n = 3 to avoid the singularity at n = 1 (although you can formally take denominator to b infinity there...).

The series is divergent for all real p. do the integral test to see this: The series converges if and only if the integral

Int [x=3 to infinity] 1/[x(ln x) (ln(ln x))^p dx

converges. This is true if the terms in the series are positive and the function of x which you choose to match the terms of the series is a monotonic decreasing function, which is indeed the case here. We start at x = 3 because for n = 2 the term ln(ln(2)) is negative so for general the theorem isn't valid. But you only need to investigate a tail of the summation starting at some arbitrary n to infinity to extablish convergence or non-convergence.

If yom change varibles y = exp(exp(x) in the integral you see that the integral is:

Int [y=ln(ln(3) to infinity] y^(p)exp(y)dy

which is divergent for all p.

correction, the ln(ln(x)) factor is in the denominator and the integral is therefore:

Int [y=ln(ln(3) to infinity] y^(-p)exp(y)dy

which is still diverges for all p.

## Similar Questions

1. ### Series

For what values of p>0 does the series a) Riemann Sum [n=1 to infinity] 1/ [n(ln n)^p] converge and for what values does it diverge?
2. ### math

for what values of x does the series converge absolutely and for what values does it converge conditionally?
3. ### Calculus

Does the series (1+sin(n))/(10^n) from summation 0 to positive infinity converge or diverge?
4. ### calculus

The sum from n=1 to infinity of cos(npi/3)/n! Does this absolutely converge, conditionally converge, or diverge?
5. ### Calculus 2

n=1 series to infinity (-5^n)/n^3 does it absolutely converge, diverge or conditionally converge. Would I be applying the ratio test?
6. ### Calculus

does the sum 3/(2^(n)-1) converge or diverge?
7. ### Calculus-PLZ HELP! =)

does the sum 3/(2^(n)-1) converge or diverge?
8. ### Calculus

does the sum e^n / n! converge or diverge?
9. ### Calculus

does the series 1/(n^2+n) converge or diverge?
10. ### Algebra 2

I need steps on how to complete this please i am so confused and lost. :( Consider the infinite geometric series x e n=1 -4(1/3) n-1. In this image, the lower limit of the summation notation is "n=1". a. Write the first four terms …

More Similar Questions