for-what-value-p-is-a-series-n-1-1-n-sin-1-n-p-convergens-

Question Number 31672 by gunawan last updated on 12/Mar/18

for what value p is a series

Missing \left or extra \right

Commented by abdo imad last updated on 12/Mar/18

s i n x = x - \frac{x^{3}}{6} + o (x^{5}) \Rightarrow s i n (\frac{1}{n}) = \frac{1}{n} - \frac{1}{6 n^{3}} + o (\frac{1}{n^{5}}) (n \to \infty)

\Rightarrow - s i n (\frac{1}{n}) = - \frac{1}{n} + \frac{1}{6 n^{3}} + o (\frac{1}{n^{5}}) \Rightarrow \frac{1}{n} - s i n (\frac{1}{n}) = \frac{1}{6 n^{3}} + o (\frac{1}{n^{5}}) \Rightarrow

{(\frac{1}{n} - s i n (\frac{1}{n}))}^{p} \sim \frac{1}{6 n^{3 p}} a n d t h e s e r i e \sum_{p ⩾ 1} \frac{1}{6 n^{3 p}} c o n v e r g e s

i f 3 p > 1 \Leftrightarrow p > \frac{1}{3} f p i s i n t e g r w e m u s t h a v e p ⩾ 1 .

Commented by gunawan last updated on 12/Mar/18

thank you Sir

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