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Question Number 203774 by Calculusboy last updated on 27/Jan/24

d e t e r m i n e w h e t h e r t h e s e r i e s i s

c o n v e r g e n t o r d i v e r g e n t

\underset{n = 1}{\sum^{\infty}} \frac{n}{\sqrt{4 n^{2} + 1}}

Answered by witcher3 last updated on 27/Jan/24

\frac{n}{\sqrt{{4 n}^{2} + 1}} ⩾ \frac{n}{\sqrt{1 + {4 n}^{2} + 4 n}} = \frac{n}{2 n + 1} = \frac{3 n}{3 (2 n + 1)} > \frac{2 n + 1}{3 (2 n + 1)}

\Rightarrow Σ \frac{n}{\sqrt{{4 n}^{2} + 1}} > Σ \frac{1}{3} Dv

Answered by Mathspace last updated on 30/Jan/24

\frac{n}{\sqrt{4 n^{2} + 1}} = \frac{n}{2 n \sqrt{1 + \frac{1}{4 n^{2}}}} = \frac{1}{2 \sqrt{1 + \frac{1}{4 n^{2}}}}

w e h a v e {(1 + x)}^{α} = 1 + α x + \frac{α (α - 1)}{2} x^{2} + o (x^{2}) \Rightarrow

\frac{1}{\sqrt{1 + x}} = {(1 + x)}^{- \frac{1}{2}}

= 1 - \frac{x}{2} + \frac{1}{2} (- \frac{1}{2}) (- \frac{1}{2} - 1) x^{2} + o (x^{2})

= 1 - \frac{x}{2} - \frac{1}{4} \times \frac{- 3}{2} x^{2} + o (x^{2})

= 1 - \frac{x}{2} + \frac{3}{8} x^{2} + o (x^{2}) s o

u_{n} = \frac{1}{2 \sqrt{1 + \frac{1}{4 n^{2}}}} = \frac{1}{2} - \frac{1}{16 n^{2}} + \frac{3}{16} \times \frac{1}{16 n^{4}} + o (\frac{1}{n^{4}})

e s t u n d e v . a s s y m t o t i q u e d e

u_{n} o n a l i m u_{n} = \frac{1}{2} \neq 0 \Rightarrow

Σ u_{n} e s t d i v e r g e n t e .