The given series is:

$x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (\text{if } x < 1)$

This is a power series, and it resembles the expansion for the natural logarithm function, specifically for $\ln(1+x)$.

The general form of the power series for $\ln(1+x)$ is:

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$

Thus, the given series can be identified as the power series for $\ln(1+x)$, valid for $x < 1$. Therefore, the sum of this infinite series is:

$\ln(1+x)$

Is this what you were looking for, or do you have additional questions?

Related Questions:

1. How does this series relate to the Taylor series expansion of $\ln(1+x)$?
2. What is the radius of convergence for this series?
3. How would this series change if $x > 1$?
4. Can you explain how to derive the series expansion for $\ln(1+x)$?
5. What happens if we truncate the series after a few terms?

Tip:

Power series expansions like this one are useful for approximating complex functions in calculus and analysis!