# Indefinite integral & series

1. Jul 9, 2011

### cal.queen92

1. The problem statement, all variables and given/known data

The problem is divided into two sections:

a) does the improper integral: 2ln(x)/x^7 (from 1 to infinity) Converge or diverge? If it converges, to what value?

b) Determine whether the series: sigma n=1 to infinity (2ln(n)/n^7) converges or diverges.

2. Relevant equations

Integration by parts?

3. The attempt at a solution

For the first part, I made the limit as c--> infinity, and took out the 2, then I simply integrated by parts where:

u = ln(x) du= 1/x dx dv= x^7 dx v = (x^8)/8

and ended up with: I = 1/4 lim c--> inf (x^8*ln(x) + (x^8)/8) from 1 to c

When I work it out, I get it Diverges, as the end result is infinity... But it's supposed to converge...

As for the second part, I assumed that they were similar where I could use the integral series test (ending up with the same result as the first part) to get the answer, but again, the answer lead to convergence....

Am I using the correct techniques?

Thanks!

2. Jul 9, 2011

### SammyS

Staff Emeritus
How did you integrate 2ln(x)/x7 ? That's the same as 2(x-7)ln(x)

I would expect the anti-derivative to have x-6 in it, not x8 !

3. Jul 9, 2011

### cal.queen92

Thank you! That was it, didn't take the proper dv -- took x^7 as oppose to 1/x^7 giving x^-7.

Perfect!