Why is the Bernoulli Number $B_1$ sometimes $+ \frac{1}{2} $?

Question

By using the recursive formula,

\begin{equation} \sum_{i=0}^{n} \binom{n+1}{i} B_i = n+1 \end{equation}

we find $B_1$ to be $\frac{1}{2}$ and not $- \frac{1}{2}$. Why is this?

Answer 1

Hans Lundmark's comment that there are different conventions in the definition of $B_n$ might bring up the question "why can there be more than one convention"?

This is the true reason (answer) to your question.

Simply put, $B_2$ through $B_n$ can be constructed from $B_2$. Therefore $B_0$ and $B_1$ can be defined however we need them to be to fit our needs, including the fact that we can simply omit both of them.

If you wish to know how I learned this on my own "from scratch", feel free to read the content (and view the images in the "spoiler tags") I have written in the spoiler of this post up until the line "I then wrote the following recursion formula from the triangle:". (Specifically, see this image. You will see that I assigned $B_1 = -1/2$ in the image, but I could have just assigned it to be "x" if I wanted to, and that wouldn't affect the values of $B_2$ through $B_n$.)

Lastly, for a video explanation and proof of my "adjusted Pascal's triangle" for the integer power sum formulas, you may also watch my YouTube video presentation.