The provided image contains a mathematical statement for a series involving alternating odd numbers. It is stated as:

$\forall n \in \mathbb{N}: \quad 1 - 3 + 5 - 7 + 9 - \cdots + (-1)^n (2n+1) = (-1)^n (n+1)$

Let's break down this expression:

1. Series Representation: This is an alternating series of odd numbers up to $(2n + 1)$. The series alternates in sign, starting with positive, and the general term is given by $(-1)^k (2k + 1)$ for $k$ ranging from 0 to $n$.

2. Sum Expression: The right side of the equation, $(-1)^n (n + 1)$, provides a compact form of the sum of this series.

Verification

To verify the equation, we can consider some small values of $n$.

• For $n = 1$: $1 - 3 = (-1)^1 \times 2 = -2$ So, the formula holds.

• For $n = 2$: $1 - 3 + 5 = 3 = (-1)^2 \times 3 = 3$ This also confirms the formula.

General Proof Strategy

A proof would involve using mathematical induction to show that the formula holds for all $n \in \mathbb{N}$.

Would you like a detailed proof using mathematical induction or have any specific questions about this series?

Here are some related questions to explore:

1. How do you derive the formula for the sum of an alternating series of odd numbers?
2. How does mathematical induction work for proving such statements?
3. Can this series be generalized for non-alternating sums?
4. What is the sum of just the positive odd numbers up to $(2n+1)$?
5. How does this series relate to other famous alternating series, like the alternating harmonic series?

Tip: For alternating series, always check the first few terms and consider induction as a powerful proof technique.