In my calc 3 class were using the divergence and integral test to determine wheter series converge or diverge. Im drawing blanks on this problem !
K / ln(k+1)
Hello, nikie1o2!
I'd use the Comparison Test . . .Converge or diverge? . $\displaystyle S \;=\;\sum^{\infty}_{k=1} \frac{k}{\ln(k+1)} $
We have: .$\displaystyle \ln(k+1) \:<\:k+1$
Then: .$\displaystyle \frac{\ln(k+1)}{k} \:<\:\frac{k+1}{k}\quad \text{ for positive }k$
Invert: .$\displaystyle \frac{k}{\ln(k+1)} \:>\:\frac{k}{k+1}$
Take sums: .$\displaystyle S \;=\;\sum^{\infty}_{k=1}\frac{k}{\ln(k+1)} \;> \;\sum^{\infty}_{k=1}\frac{k}{k+1} $
Now: .$\displaystyle \sum^{\infty}_{k=1}\frac{k}{k+1} \;>\;\sum^{\infty}_{k=1}\frac{1}{k+1} \;=\;\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \hdots$ divergent Harmonic Series
Hence: .$\displaystyle S \:>\:\text{(divergent series)}$
Therefore: .$\displaystyle S$ diverges.