Let $\map f x = e^{\tan x}$. Then: {{begin-eqn}} {{eqn | l = \frac \d {\d x} \map f x       | r = \sec^2 x \, e^{\tan x}       | c = Chain Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \frac {\d^2} {\d x^2} \map f x       | r = \sec^2 x \frac \d {\d x} e^{\tan x} + e^{\tan x} \frac \d {\d x} \sec^2 x       | c = Product Rule }} {{eqn | r = \sec^4 x \, e^{\tan x} + e^{\tan x} 2 \sec^2 x \tan x       | c =  }} {{eqn | r = \paren {\sec^2 x + 2 \tan x} \sec^2 x \, e^{\tan x}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d^3} {\d x^3} \map f x       | r = \paren {\sec^2 x + 2 \tan x} \frac \d {\d x} \sec^2 x \, e^{\tan x} + \sec^2 x \, e^{\tan x} \frac \d {\d x} \paren {\sec^2 x + 2 \tan x}       | c = Product Rule }} {{eqn | r = \paren {\sec^2 x + 2 \tan x} \paren {\sec^2 x + 2 \tan x} \sec^2 x \, e^{\tan x} + \sec^2 x \, e^{\tan x} \paren {2 \sec^2 x \tan x + 2 \sec^2 x}       | c =  }} {{eqn | r = \paren {\paren {\sec^2 x + 2 \tan x}^2 + 2 \sec^2 x \tan x + 2 \sec^2 x} \sec^2 x \, e^{\tan x}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d^4} {\d x^4} \map f x       | r = \paren {\paren {\sec^2 x + 2 \tan x}^2 + 2 \sec^2 x \tan x + 2 \sec^2 x} \frac \d {\d x} \sec^2 x \, e^{\tan x}       | c = Product Rule }} {{eqn | o =        | ro= +       | r = \sec^2 x \, e^{\tan x} \frac \d {\d x} \paren {\paren {\sec^2 x + 2 \tan x}^2 + 2 \sec^2 x \tan x + 2 \sec^2 x}       | c =  }} {{eqn | r = \paren {\paren {\sec^2 x + 2 \tan x}^2 + 2 \sec^2 x \tan x + 2 \sec^2 x} \paren {\sec^2 x + 2 \tan x} \sec^2 x \, e^{\tan x}       | c =  }} {{eqn | o =        | ro= +       | r = \sec^2 x \, e^{\tan x} \paren {2 \paren {\sec^2 x + 2 \tan x} \paren {2 \sec^2 x \tan x + 2 \sec^2 x} + 4 \sec^2 x \tan^2 x + 2 \sec^4 x + 4 \sec^2 x \tan x}       | c =  }} {{end-eqn}} By definition of Taylor series: :$\map f x \sim \displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ This is to be expanded about $\xi = 0$. Note that $\tan 0 = 0$ and $\sec 0 = 1$. Thus: {{begin-eqn}} {{eqn | l = \map f 0       | r = e^0       | c =  }} {{eqn | r = 1       | c =  }} {{eqn | l = \valueat {\frac \d {\d x} \map f x} {x \mathop = 0}       | r = \sec^2 0 \, e^{\tan 0}       | c =  }} {{eqn | r = 1 \times e^0       | c =  }} {{eqn | r = 1       | c =  }} {{eqn | l = \valueat {\frac {\d^2} {\d x^2} \map f x} {x \mathop = 0}       | r = \paren {\sec^2 0 + 2 \tan 0} \sec^2 0 \, e^{\tan 0}       | c =  }} {{eqn | r = \paren {1 + 0} 1 \times e^0       | c =  }} {{eqn | r = 1       | c =  }} {{eqn | l = \valueat {\frac {\d^3} {\d x^3} \map f x} {x \mathop = 0}       | r = \paren {\paren {\sec^2 0 + 2 \tan 0}^2 + 2 \sec^2 0 \tan 0 + 2 \sec^2 0} \sec^2 0 \, e^{\tan 0}       | c =  }} {{eqn | r = \paren {\paren {1 + 0}^2 + 2 \times 1 \times 0 + 2} 1 \times e^0       | c =  }} {{eqn | r = 3       | c =  }} {{eqn | l = \valueat {\frac {\d^4} {\d x^4} \map f x} {x \mathop = 0}       | r = \paren {\paren {\sec^2 0 + 2 \tan 0}^2 + 2 \sec^2 0 \tan 0 + 2 \sec^2 0} \paren {\sec^2 0 + 2 \tan 0} \sec^2 0 \, e^{\tan 0}       | c =  }} {{eqn | o =        | ro= +       | r = \sec^2 0 \, e^{\tan 0} \paren {2 \paren {\sec^2 0 + 2 \tan 0} \paren {2 \sec^2 0 \tan 0 + 2 \sec^2 0} + 4 \sec^2 0 \tan^2 0 + 2 \sec^4 0 + 4 \sec^2 0 \tan 0}       | c =  }} {{eqn | r = \paren {\paren {1 + 0}^2 + 0 + 2} \paren {1 + 0} + \paren {2 \paren {1 + 0} \paren {0 + 2} + 0 + 2 + 0}       | c =  }} {{eqn | r = 9       | c =  }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi       | c =  }} {{eqn | r = \paren {1 \times 1} + \paren {x \times 1} + \paren {\dfrac {x^2} {2!} \times 1} + \paren {\dfrac {x^3} {3!} \times 3} + \paren {\dfrac {x^4} {4!} \times 9} + \cdots       | c =  }} {{eqn | r = 1 + x + \dfrac {x^2} 2 + \dfrac {x^3} 2 + \dfrac {3 x^4} 8 + \cdots       | c =  }} {{end-eqn}} {{qed}}