Find the Taylor Series of F(x) = $\frac { 1 } { 1 + x }$

question 102

True/False

Find the Taylor series of f(x) = $\frac { 1 } { 1 + x }$ for $| x |$ < 1 then multiply the series by x to obtain a series expansion of $\frac { x } { x + 1 }$ , and then use these two series to obtain a series expansion of $\frac { 1 - x } { 1 + x }$ for $| x |$ < 1.
Is $\frac { 1 - x } { 1 + x } = 1 + 2 x + 2 x ^ { 2 } + 2 x ^ { 3 } + 2 x ^ { 4 } + \ldots$ the correct expansion?

---

Definitions:

Right from Wrong

The ability to distinguish ethical behavior from unethical behavior based on moral principles.

Ethical Justifications

The reasoning provided to explain why a particular action or decision is morally right.

Self-Serving Decisions

Choices made with the primary aim of benefiting oneself, often at the expense of others or ethical considerations.

Others' Rights

The acknowledgment and respect for the rights and freedoms of other individuals, essential for maintaining ethical and harmonious interactions.