calculate the value of Σ_(n=0) ^(1947) (1/(2^n +(√2^(1947) )))


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Question Number 62669 by Sayantan chakraborty last updated on 24/Jun/19

$c a l c u l a t e t h e v a l u e o f \underset{n = 0}{\sum^{1947}} \frac{1}{2^{n} + \sqrt{2^{1947}}}$
	Answered by tanmay last updated on 24/Jun/19
	$a=(√2^(1947) ) S=(1/(2^0 +a))+(1/(2^1 +a))+(1/(2^2 +a))+(1/(2^3 +a))+...+(1/(2^(1947) +a)) ((2+a)/2)≥(2×a)^(1/2) 2+a≥2×(2a)^(1/2) (1/(2+a))≤(1/(2×(2a)^(1/2) )) (1/(2+a))≤(1/(2(√a) ×(2)^(1/2) )) S≤(1/(2(√a)))[(1/((2^0 )^(1/2) ))+(1/((2^1 )^(1/2) ))+(1/((2^2 )^(1/2) ))+...+(1/((2^(1947) )^(1/2) ))] S≤(1/(2(√a)))[(1/(((√(2 )) )^0 ))+(1/(((√2) )^1 ))+(1/(((√2) )^2 ))+...+(1/(((√2) )^(1947) ))] S≤(1/(2(√a)))×(((1/(((√2) )^0 )){1−((1/((√2) )))^(1948) })/(1−((1/(√2))))) [using ((A(1−r^n ))/(1−r))] S≤(1/(2(√a)))×(((√2)×(1−(1/2^(974) )))/((√2) −1)) S≤(1/((2−(√2) ))) ×((1−(1/2^(974) ))/(((√2^(1947) ) )^(1/2) ))≈(1/((2−(√2) )))×(1/(((√2^(1947) ) )^(1/2) )) i have found approximate value not exact value so please cheack$
	$a = \sqrt{2^{1947}}$ $S = \frac{1}{2^{0} + a} + \frac{1}{2^{1} + a} + \frac{1}{2^{2} + a} + \frac{1}{2^{3} + a} + . . . + \frac{1}{2^{1947} + a}$ $\frac{2 + a}{2} ⩾ {(2 \times a)}^{\frac{1}{2}}$ $2 + a ⩾ 2 \times {(2 a)}^{\frac{1}{2}}$ $\frac{1}{2 + a} ⩽ \frac{1}{2 \times {(2 a)}^{\frac{1}{2}}}$ $\frac{1}{2 + a} ⩽ \frac{1}{2 \sqrt{a} \times {(2)}^{\frac{1}{2}}}$ $S ⩽ \frac{1}{2 \sqrt{a}} [\frac{1}{{(2^{0})}^{\frac{1}{2}}} + \frac{1}{{(2^{1})}^{\frac{1}{2}}} + \frac{1}{{(2^{2})}^{\frac{1}{2}}} + . . . + \frac{1}{{(2^{1947})}^{\frac{1}{2}}}]$ $S ⩽ \frac{1}{2 \sqrt{a}} [\frac{1}{{(\sqrt{2})}^{0}} + \frac{1}{{(\sqrt{2})}^{1}} + \frac{1}{{(\sqrt{2})}^{2}} + . . . + \frac{1}{{(\sqrt{2})}^{1947}}]$ $S ⩽ \frac{1}{2 \sqrt{a}} \times \frac{\frac{1}{{(\sqrt{2})}^{0}} {1 - {(\frac{1}{\sqrt{2}})}^{1948}}}{1 - (\frac{1}{\sqrt{2}})} [u s i n g \frac{A (1 - r^{n})}{1 - r}]$ $S ⩽ \frac{1}{2 \sqrt{a}} \times \frac{\sqrt{2} \times (1 - \frac{1}{2^{974}})}{\sqrt{2} - 1}$ $S ⩽ \frac{1}{(2 - \sqrt{2})} \times \frac{1 - \frac{1}{2^{974}}}{{(\sqrt{2^{1947}})}^{\frac{1}{2}}} \approx \frac{1}{(2 - \sqrt{2})} \times \frac{1}{{(\sqrt{2^{1947}})}^{\frac{1}{2}}}$ $i h a v e f o u n d a p p r o x i m a t e v a l u e n o t$ $e x a c t v a l u e s o p l e a s e c h e a c k$
	Commented by Sayantan chakraborty last updated on 25/Jun/19

	$a n s w e r i s \frac{487}{\sqrt{} 2^{1945}} .$
	Answered by ajfour last updated on 30/Jun/19

	$S = \underset{n = 0}{\sum^{2 N}} \frac{1}{2^{n} + 2^{N}} (w h e r e 2 N = 1947)$ $= \frac{1}{2^{N}} (\underset{n = 0}{\sum^{2 N}} \frac{1}{1 + 2^{n - N}})$ $2^{N} S = \frac{2^{N}}{2^{N} + 1} + \frac{2^{N}}{2^{N} + 2} + \frac{2^{N}}{2^{N} + 2^{2}} + . . .$ $. . . . . . + \frac{2}{2 + 1} + \frac{1}{1 + 1} + \frac{1}{1 + 2} + \frac{1}{1 + 2^{2}} + . .$ $. . . . . + \frac{2^{2}}{2^{2} + 2^{N} +} \frac{2}{2 + 2^{N}} + \frac{1}{1 + 2^{N}} + . . . .$ $\Rightarrow 2^{N} S = \frac{1}{2} + N$ $S = \frac{2 N + 1}{2 \sqrt{2^{2 N}}}$ $\Rightarrow S = \frac{1948}{2 (\sqrt{} 2^{1947})} = \frac{487}{\sqrt{} 2^{1945}} .$
	Commented by Sayantan chakraborty last updated on 30/Jun/19

	$Y e s i t i s c o r r e c t s o l u t i o n .$
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