Question

Use a series you know to show that $\sum _{n=0}^{\mathrm{\infty }}\frac{(1+i\pi {)}^n}{n!}=-e$.

Short answer

$\sum _{n=0}^{fty}\frac{(1+i\pi {)}^n}{n!}=-e$.

Step-by-step solution

- Identify the Series

Recognize that the given sum $\sum _{n=0}^{fty}\frac{(1+i\pi {)}^n}{n!}$ represents an exponential series. In general, the exponential series is given by $e^x=\sum _{n=0}^{fty}\frac{x^n}{n!}$.

- Compare with the Exponential Series

Understand that $\sum _{n=0}^{fty}\frac{(1+i\pi {)}^n}{n!}$ matches the form of the exponential series $e^x$. Here, $x=1+i\pi$.

- Apply the Exponential Series Formula

Using the exponential series formula, substitute $x=1+i\pi$ into the formula to obtain $e^{1+i\pi }$.

- Simplify the Exponential Expression

Recall Euler's formula, which states $e^{ix}=\cos (x)+i\sin (x)$. Therefore, $e^{i\pi }=-1$. Substituting this into the expression, we get $e^{1+i\pi }=e{e}^{i\pi }=e(-1)=-e$.

Key concepts

Euler's Formula

Euler's formula is a stunning result in mathematics, revealing a deep connection between complex exponentials and trigonometric functions. Euler's formula states that for any real number x, \br


Euler's formula is often used to solve problems involving complex numbers and trigonometry, making it a vital tool in various fields such as electrical engineering and physics. In the context of the given exercise, Euler's formula helps us simplify the complex exponential expression. For example, in Step 4 of the solution, we use the fact that $e^{i\pi }=\cos (\pi )+i\sin (\pi )$. Since $\cos (\pi )=-1$ and $\sin (\pi )=0$, we get $e^{i\pi }=-1$. This result is crucial for simplifying and solving the given series. Understanding Euler's formula allows students to handle a broader range of problems involving complex numbers and exponential functions.

Complex Numbers

Complex numbers might seem daunting, but let's simplify the concept. A complex number is of the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying $i^2=-1$. This makes complex numbers an extension of the real numbers. To grasp them better: