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Calculus Level 5

a , b are coprime and c, d are coprime and ln (e)=1 2 1 ! + 4 2 ! + 6 4 ! + 8 6 ! + . . . . . + ∞ = a e b + c d e \frac{2}{1!}+\frac{4}{2!}+\frac{6}{4!}+\frac{8}{6!}+.....+\infty=\frac{ae}{b}+\frac{c}{de}

Find the value of a b 3 d c 3 ab^{3}dc^{3}


The answer is 48.

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Milun Moghe by Milun Moghe , 25, India

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1 solution

Milun Moghe
Jan 9, 2014

s i n ( x ) + x c o s ( x ) = 2 x − x 3 ( 1 2 ! + 1 3 ! ) + x 5 ( 1 4 ! + 1 5 ! ) − x 7 ( 1 6 ! + 1 7 ! ) + . . . . . . . . . sin(x)+xcos(x)=2x-x^{3}(\frac{1}{2!}+\frac{1}{3!})+x^{5}(\frac{1}{4!}+\frac{1}{5!})-x^{7}(\frac{1}{6!}+\frac{1}{7!})+......... 2 c o s ( x ) − x s i n ( x ) = 2 − 3 x 2 ( 1 2 ! + 1 3 ! ) + 5 x 4 ( 1 4 ! + 1 5 ! ) − . . . . . . . . 2cos(x)-xsin(x)=2-3x^{2}(\frac{1}{2!}+\frac{1}{3!})+5x^{4}(\frac{1}{4!}+\frac{1}{5!})-........ 2 c o s ( i ) − i s i n ( i ) = 2 − 3 i 2 ( 1 2 ! + 1 3 ! ) + 5 i 4 ( 1 4 ! + 1 5 ! ) − . . . . . . . . 2cos(i)-isin(i)=2-3i^{2}(\frac{1}{2!}+\frac{1}{3!})+5i^{4}(\frac{1}{4!}+\frac{1}{5!})-........ using hyperbolic functions or formulae of sin and cos 1 2 e + 3 e 2 \frac{1}{2e}+\frac{3e}{2}

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