Knowlpedia

Identity

Telescoping sum

Cancellation of successive differences in a finite sum and the additional limit needed for an infinite sum.

Core idea

For b0,…,bnb_0,\ldots,b_n in an additive , the telescoping identity is

∑j=1n(bj−bj−1)=bn−b0.\sum_{j=1}^{n}(b_j-b_{j-1})=b_n-b_0.

To prove it, expand the finite sum: every intermediate bjb_j appears once with each sign and cancels.

Infinite limits

If the bjb_j lie in a normed vector space and converge to bb, the identity for partial sums gives

∑j=1∞(bj−bj−1)=b−b0.\sum_{j=1}^{\infty}(b_j-b_{j-1})=b-b_0.

This establishes convergence in that norm. It need not establish absolute convergence of the series of norms. Without convergence of bnb_n, the finite cancellation identity alone gives no infinite sum.