study the convergence of Σ_(n=0) ^∞ sin(π(√(4n^2 +1)))


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Question Number 52670 by maxmathsup by imad last updated on 11/Jan/19

$s t u d y t h e c o n v e r g e n c e o f \sum_{n = 0}^{\infty} s i n (π \sqrt{4 n^{2} + 1})$
Commented by maxmathsup by imad last updated on 12/Jan/19

$w e h a v e π \sqrt{4 n^{2} + 1} = π \sqrt{4 n^{2} (1 + \frac{1}{4 n^{2}}}) = 2 n π \sqrt{1 + \frac{1}{4 n^{2}}} b u t w e h a v e$ $\sqrt{1 + x} = {(1 + x)}^{\frac{1}{2}} = 1 + \frac{x}{2} + \frac{\frac{1}{2} (\frac{1}{2} - 1)}{2} x^{2} + o (x^{3}) (x \to 0) \Rightarrow$ $= 1 + \frac{x}{2} - \frac{1}{8} x^{2} + o (x^{3}) \Rightarrow \sqrt{1 + \frac{1}{4 n^{2}}} = 1 + \frac{1}{8 n^{2}} - \frac{1}{8.16 n^{4}} + o (\frac{1}{n^{6}}) \Rightarrow$ $2 n π \sqrt{1 + \frac{1}{4 n^{2}}} = 2 n π + \frac{π}{4 n} - \frac{π}{32 n^{3}} + o (\frac{1}{n^{5}}) \Rightarrow s i n (π \sqrt{4 n^{2} + 1}) \sim s i n (\frac{π}{4 n}) \sim \frac{π}{4 n}$ $b u t Σ \frac{π}{4 n} d i v e r g e s \Rightarrow Σ s i n (π \sqrt{4 n^{2} + 1}) d i v e r g e s .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 11/Jan/19
	$((4n^2 +1)/2)≥(√(4n^2 ×1)) 4n^2 +1≥4n (√(4n^2 +1)) ≥(√(4n)) sin(π(√(4n^2 +1)) )≈sin(π×2(√n) ) sin((√n) ×2π) 1)now when n is perfect square sin((√n) ×2π)=0 2)(√n) =I+f I=integer f=fractional part sin{(I+f)2π} =sin{I×2π+f×2π} =sin(f×2π) 1≥sin(2πf)≥−1 so Σ_(n=0) ^∞ sin(π(√(4n^2 +1)) ) does not converge Σ_(n=0) ^∞ sin(π(√(4n^2 +1)) ) oscilates between ± ∞ i have tried to solve ...pls check is the logic true...$
	$\frac{4 n^{2} + 1}{2} ⩾ \sqrt{4 n^{2} \times 1}$ $4 n^{2} + 1 ⩾ 4 n$ $\sqrt{4 n^{2} + 1} ⩾ \sqrt{4 n}$ $s i n (π \sqrt{4 n^{2} + 1}) \approx s i n (π \times 2 \sqrt{n})$ $s i n (\sqrt{n} \times 2 π)$ $1) n o w w h e n n i s p e r f e c t s q u a r e$ $s i n (\sqrt{n} \times 2 π) = 0$ $2) \sqrt{n} = I + f I = i n t e g e r$ $f = f r a c t i o n a l p a r t$ $s i n {(I + f) 2 π}$ $= s i n {I \times 2 π + f \times 2 π}$ $= s i n (f \times 2 π)$ $1 ⩾ s i n (2 π f) ⩾ - 1$ $s o \underset{n = 0}{\sum^{\infty}} s i n (π \sqrt{4 n^{2} + 1}) d o e s n o t c o n v e r g e$ $\underset{n = 0}{\sum^{\infty}} s i n (π \sqrt{4 n^{2} + 1}) o s c i l a t e s b e t w e e n \pm \infty$ $i h a v e t r i e d t o s o l v e . . . p l s c h e c k i s t h e l o g i c$ $t r u e . . .$
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