Alternating Summation of Ones

Here is a pretty common high school problem that shows the weirdness that happen with infinite series.

\text{What is the result of } 1-1+1-1+1\ldots\text{?}

We want to know what the value is so set equal to some unknown, like $\text{X}$ and notice that it contains itself.

\begin{align*} & \text{X} = 1 \overbrace{ - 1 + 1 - 1 + 1-1\ldots}^{-\text{X}}\\ & \text{X} = 1 -\text{X}\\ & 2\text{X} = 1 \rightarrow \boxed{\text{X} = 1/2} \end{align*}

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\text{What is the result of } 1-1+1-1+1\ldots\text{?}

We want to know what the value is so set equal to some unknown, like $\text{X}$ and notice that it contains itself.

\begin{align*} & \text{X} = 1 \overbrace{ - 1 + 1 - 1 + 1-1\ldots}^{-\text{X}}\\ & \text{X} = 1 -\text{X}\\ & 2\text{X} = 1 \rightarrow \boxed{\text{X} = 1/2} \end{align*}