(Solved): - Find the Taylor series of ex,cos(x),sin(x),1x1 - Verify Euler's identity. - Find the Ta ...

- Find the Taylor series of $$e^x,\cos (x),\sin (x),\frac{1}{1?x}$$ - Verify Euler's identity. - Find the Taylor series of other functions via substitution, integration, differentiation, etc. Problem 01: Using the Taylor series of $$\frac{1}{1?x}$$, find the Taylor series, centered at 0, of each of the following functions: a) $$\frac{1}{1?2x}$$ b) $$\frac{x}{1?x}$$ c) $$\frac{1}{(1?x{)}^2}$$ (Hint: Squaring a series is not practical, since it is impossible to distribute a product containing an infinite number of terms. However, isn't $$\frac{1}{(1?x{)}^2}$$ the derivative of the function $$\frac{1}{1?x}$$ ?) d) $$\frac{1}{1+x}$$ e) $$\frac{1}{1+x^2}$$ f) $$\arctan (x)$$ (Hint: Isn't $$\arctan (x)$$ an antiderivative of $$\frac{1}{1+x^2}$$ ?)