but now you're on board the spaceship #1431838547, and the captain is asking to solve tricky "ZP" (Z- Planet) problem (you know what it means when captain is "asking"... that's an order): Find the sum of the series: (-1)2 25 -3 n = 0 (2n + 1)! for Z-Planet conditions! surprising way. !!! The challenge is some of the Maclaurin Series for very basic functions are twisted on Planet Z in a on Earth (- 1)" 2n +1 n! , sinc E (- 1)"x2 COS M n=0 (2n + 1) ! n=0 (2n) ! 1 - T 1 n=0 1 + x C (-1)"r" , arctanr = ( - 1)" x 2 +1 (- 1) \ - 1 M n=0 7 =0 2n + 1 In(1 + x) = n =1 n but on Z-planet Va er (- 1)" Vx 2n + 1 sin * = (- 1)" Vx2 n! cos x n=0 n=0 (2n + 1)! n=0 (2n)! 1 00 (- 1)" Va2+1 1 - T n= 0 1+x E ( - 1)" va" , arctanI 7 = 0 n =0 2n + 1 (- 1)2-1 Va" In(1 + x) n n =1 ( all the other rules have not been changed). So, for Z-Planet conditions the sum of the series (- 1)"250-3 is : n = 0 (2n + 1) !but now you're on board the spaceship #1884047532, and the captain is asking to solve tricky "ZP" (Z- Planet) problem (you know what it means when captain is "asking"... that's an order): Find the sum of the series: E (- 2) 12 2on+3 o (2n + 1) . 27-1 for Z-Planet conditions! !!! The challenge is some of the standard Taylor/Maclaurin Series are twisted on Planet Z in a surprising way. on Earth (- 1)" x 2n + 1 , sinc = (- 1)" x2 M 8 1 n=0 n! COST n=0 (2n + 1)! n=0 (2n) ! 1 - n=0 (- 1)"x" , arctanI 72 - 1 1 +r E (- 1)" x2+1 7 =0 2n + 1 In(1 + x) (-1)' n=0 n = 1 n but on Z-planet 8 PI (- 1)" a2+1 0o sin c n! Cos n=0 n=0 (2n + 1)! n =0 (2n) ! 1 Van 1 - C ( - 1)" Vx" , arctanr = (- 1)" Va2+1 7=0 1 +x n=0 n=0 2n + 1 (- 1)2-1 Van In(1 + x) = n =1 n ( all the other rules have not been changed). So, for Z-Planet conditions the sum of the series ( - 2) " on + 3 M 8 is : n=0 (2n + 1) . 20-1but now you're on board the spaceship #1788760332, and the captain is asking to solve tricky "ZP" (Z- Planet) problem: The Taylor series for f(x) = x at 1 is n =0 Find the first few coefficients for Z-Planet conditions! CO C1 = C2 C3 C4 !!! The challenge is some laws of nature are twisted on Planet Z in a surprising way, in particular, the d n - easiest differentiation rule on Earth, = nx" dx d -1 on Z planet appears as n d const = 0 , and the other rules have not been changed) dx