Step 1: Understanding the Concept:
This problem involves the expansion of the logarithmic series \( \log(1+x) \). : Key Formula or Approach:
The standard series expansion is:
\[ \log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \]
For \( x = 1 \):
\[ \log 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \]
Step 2: Detailed Explanation:
The given summation is:
\[ S = \sum_{n=1}^{\infty} \frac{(-1)^n}{n+1} \]
Expand the sum:
\[ S = \frac{-1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots \]
Compare this with the expansion for \( \log 2 \):
\[ \log 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \]
Notice that:
\[ \log 2 = 1 + \left( -\frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \right) \]
\[ \log 2 = 1 + S \]
Rearranging for \( S \):
\[ S = \log 2 - 1 \]. Step 3: Final Answer:
The value of the sum is \( \log 2 - 1 \).