Converge or Diverge?
I'm not sure how to go about solving these...help?
1. Sigma (Sqrt(2n+1) - Sqrt(2n))
n = 1, and goes to infinity
1. Sigma (1/(Sqrt(n)ln (n)))
n= 2, and goes to infinity
thanks.
1.Consider the sequence of partial sums for the first one. Do you notice a pattern?
2.$\displaystyle \sum_{n=2}^{\infty}\frac{1}{\sqrt{n}\ln(n)}>\sum_{ n=2}^{\infty}\frac{1}{n\ln(n)}$
Now consider that if $\displaystyle a_n=\frac{1}{n\ln(n)}$, then $\displaystyle \forall{n}\in[2,\infty)~a_n>0\wedge{a_n\in\mathcal{C}}\wedge{a_n \in\downarrow}$
So the integral test applies, and note that $\displaystyle \int_2^{\infty}\frac{dx}{x\ln(x)}$ diverges.