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Question Number 170199 by Mastermind last updated on 18/May/22

F i n d t h e f i r s t f o u r t e r m s o f t h e s e r i e s

f o r e^{x} s i n h x

M a s t e r m i n d

Answered by floor(10²Eta[1]) last updated on 18/May/22

sinhx = \frac{e^{x} - e^{- x}}{2}

∴ e^{x} sinhx = \frac{e^{2 x} - 1}{2} = f (x)

f (0) = 0

f^{'} (x) = e^{2 x} \Rightarrow f^{'} (0) = 1

f^{″} (x) = {2 e}^{2 x} \Rightarrow f^{″} (0) = 2

f^{‴} (x) = {4 e}^{2 x} \Rightarrow f^{‴} (0) = 4

\Rightarrow f^{(n)} (x) = 2^{n - 1} e^{2 x} \Rightarrow f^{(n)} (0) = 2^{n - 1}, n > 0

f (x) = \underset{n = 0}{\sum^{\infty}} \frac{f^{(n)} (0) x^{n}}{n!} = x + \frac{{2 x}^{2}}{2!} + \frac{{4 x}^{3}}{3!} + \frac{{8 x}^{4}}{4!} + \dots + \frac{2^{k - 1} x^{k}}{k!} + \dots

Commented by Mastermind last updated on 19/May/22

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