Infinite Telescoping Sum: $\sum_{i=1}^{\infty} (X_i - X_{i-1})=$?

Question

Let $(X_i)_{i \geq 0}$ be any countable sequence of numbers and suppose that a limit exists, so $$\lim_{i \rightarrow \infty} X_i = x.$$

Consider $\sum_{i=1}^{\infty} (X_i - X_{i-1})$.

Is this telescoping sum given by $x-X_0$ or by $-X_0$ only?

Answer 1

You have, for $N \geq1$, $$ \sum_{i=1}^N (X_i - X_{i-1})=X_N-X_0 $$ then letting $N \to \infty$ gives

$$ \sum_{i=1}^{\infty} (X_i - X_{i-1})=x-X_0 $$

since $\lim_{i \rightarrow \infty} X_i = x.$