find the greatest coeeficient of (1+(x/2))^(−2) i have applied Q.125697 but i got two negative values of n after i have made k=2n for positive coefficient. I believe it should have a greatest coeeficient since ((1/2))^k decreases as as k increases thanks what do i do?


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Question Number 125881 by aurpeyz last updated on 14/Dec/20

$f i n d t h e g r e a t e s t c o e e f i c i e n t o f$ ${(1 + \frac{x}{2})}^{- 2}$ $i h a v e a p p l i e d Q . 125697 b u t i g o t t w o$ $n e g a t i v e v a l u e s o f n a f t e r i h a v e$ $m a d e k = 2 n f o r p o s i t i v e c o e f f i c i e n t .$ $I b e l i e v e i t s h o u l d h a v e a g r e a t e s t$ $c o e e f i c i e n t s i n c e {(\frac{1}{2})}^{k} d e c r e a s e s a s$ $a s k i n c r e a s e s$ $t h a n k s$ $w h a t d o i d o ?$
Commented by mr W last updated on 14/Dec/20

$y e s, t h e r e i s a m a x i m u m (a s w e l l a s$ $m i n i m u m) c o e f f i c i e n t .$ ${(1 + \frac{x}{2})}^{- 2} = \underset{k = 0}{\sum^{\infty}} {(- 1)}^{k} C_{1}^{k + 1} \frac{1}{2^{k}} x^{k}$ $a_{2 n} = \frac{1}{2^{2 n}} C_{1}^{2 n + 1} = \frac{2 n + 1}{2^{2 n}}$ $a_{2 n + 1} = - \frac{1}{2^{2 n + 1}} C_{1}^{2 n + 2} = - \frac{n + 1}{2^{2 n}}$ $\frac{2 n + 1}{2^{2 n}} > \frac{2 (n + 1) + 1}{2^{2 (n + 1)}}$ $4 (2 n + 1) > 2 n + 3$ $n > - \frac{1}{6}$ $\Rightarrow n_{m i n} = 0$ $\Rightarrow a_{0} i s t h e l a r g e s t c o e f f i c i e n t$ $\Rightarrow a_{0} = 1$
Commented by aurpeyz last updated on 15/Dec/20

$w o w . t h a t s n i c e . t h a n k s a l o t .$ $i c a n n o w s o l v e a l l p r o b l e m s o n i t$
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