let-u-n-cos-pi-n-2-n-1-find-nature-of-u-n-

Question Number 32037 by abdo imad last updated on 18/Mar/18

l e t u_{n} = c o s (π \sqrt{n^{2} + n + 1}) f i n d n a t u r e o f Σ u_{n} .

Commented by abdo imad last updated on 22/Mar/18

w e h a v e π \sqrt{n^{2} + n + 1} = π n \sqrt{1 + \frac{1}{n} + \frac{1}{n^{2}}}

\sim n π (1 + \frac{1}{2} (\frac{1}{n} + \frac{1}{n^{2}}) = n π + \frac{π}{2} + \frac{π}{n} \Rightarrow

c o s (π \sqrt{n^{2} + n + 1}) \sim c o s (n π + \frac{π}{2} + \frac{π}{n}) = - s i n (n π + \frac{π}{n})

= {(- 1)}^{n - 1} c o s (\frac{π}{n}) b u t w e h a v e c o s (\frac{π}{n}) \sim 1 - \frac{π^{2}}{n^{2}} \Rightarrow

u_{n} \sim {(- 1)}^{n - 1} (1 - \frac{π^{2}}{n^{2}}) b u t Σ {(- 1)}^{n} d i v e r g e s \Rightarrow Σ u_{n} d i v e r g e s