Question

Help with integral criterion

Answer 1

You have to post the question

Answer 2

I shall show that \[\sum_{n=2}^{∞}\frac{ \ln(n)^p }{ n }\]
is convergent if p <-1 and divergent if p> -1.

I have to use the integral criterion. So before I can use the integral criterion I have to show that the series is positive, continuous and decreasing.

Can somebody please help me?

Answer 3

Dang, I could never do that......

Answer 4

show the derivative of (ln(x)^p)/x has is always negative for the decreasing part

Answer 5

The derivative of (ln(x)^p)/x is \[\frac{ p*\ln(x)^p }{ x^2 }-\frac{ \ln(x)^p }{ x^2 }=\frac{ \ln(x)^p*(p-\ln(x) }{ x^2*\ln(x) }\] How can I show that? @amistre64

Answer 6

I guess u need to find that not derivative

Answer 7

is that part of the integral criterion tho?

Answer 8

and is that ln^p or ln(x^p)?

\[\frac{ \ln(x)^p }{ x }\]
\[\frac{ x\frac{x^{p-1}}{x^p}-ln(x)^p}{x^2}\]
\[\frac{ xx^{-1}-ln(x)^p}{x^2}\]
\[\frac{ -ln(x)^p}{x^2}\]