= Telescoping sum
{title2=$\sum_{k=1}^{n}(a_k-a_{k-1})=a_n-a_0$}
A <telescoping sum> cancels intermediate <sequence terms>, leaving <boundary terms>. For a finite <sequence> of <real numbers> $a_0,\ldots,a_n$, the displayed identity follows because every $a_1,\ldots,a_{n-1}$ occurs once with each sign. Applying it to an infinite <series> also requires checking the <limit> of the <boundary terms>; finite cancellation alone does not prove that the <series> is a <convergent series>.<Calculus Volume 2 by Gilbert Strang and others/5.2 Infinite Series>[Calculus Volume 2, §5.2 Infinite Series: Telescoping Series definition and successive partial-sum identities]{ref}
For $n\geq1$, expanding the finite <sum> gives
$$
\sum_{k=1}^n(a_k-a_{k-1})
=(a_1-a_0)+(a_2-a_1)+\cdots+(a_n-a_{n-1})
=a_n-a_0.
$$
No <monotone sequence> assumption is required. Consider the original <sequence> $4,9,2,7,5$. Its successive <finite differences> are $5,-7,5,-2$, so
$$
5-7+5-2=1=5-4.
$$
The large intermediate rise and fall disappear from the net change. This is useful whenever one knows successive changes rather than every absolute value: adding the changes reconstructs the final value from the initial one. It also checks a cumulative-update calculation without requiring all increments to be positive.
\Image[/telescoping-sum-signed-cancellation-ledger.png]
{title=Opposite signs cancel the intermediate <sequence terms>, leaving the final value minus the initial value}
{description=Original nonmonotone <sequence> $4,9,2,7,5$. Linked green entries have the same magnitude and opposite signs; the unmatched entries are $+5$ and $-4$. The increments are $5,-7,5,-2$, whose <sum> is one.}
A <finite difference> $\Delta a_{k-1}=a_k-a_{k-1}$ measures one discrete change. The identity $\sum_{k=1}^n\Delta a_{k-1}=a_n-a_0$ is the discrete counterpart of the <fundamental theorem of calculus>: accumulation of changes returns the endpoint difference. The related <summation by parts> transfers a <finite difference> from one factor of a product to another, with explicit <boundary terms>.
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