It Can Be Shown That the Maclaurin Series For $e^{z}$ $\cos z$

question 47

Essay

It can be shown that the Maclaurin series for $e^{z}$ , $\cos z$ and $\sin z$ converge for all values of z in the complex numbers, just as they do for all values of x in the real numbers.
a)Write down and simplify the Maclaurin series for $e^{i x}$ .
b)Write down the Maclaurin series for $\cos x$ and $i \sin x$ c)Use the series you found in parts a)and b)to show that $e^{i x}=\cos x+i \sin x$ .(This is one of several formulas called "Euler's Formula.")
d)Find the value of $7\left(e^{i s}+1\right)$ .

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Definitions:

Standard Deviation

A measure of the dispersion or variability of a set of data points or investment returns, indicating the degree of risk or volatility.

Risk-Free Asset

An investment with a guaranteed return and no risk of financial loss, typically represented by government bonds.

Capital Allocation Line

A line that graphically represents the risk-versus-return profile of risky assets, and indicates the optimal portfolio of risky and risk-free assets.

Risk-Free Asset

An investment with a guaranteed return and no risk of financial loss, often represented by government bonds.