indexshifting

Indexshifting is a technique in mathematics used to rearrange sums or series by changing the indexing variable. It involves substituting a new index, typically m = n + k, and adjusting the summation limits accordingly. This reindexing helps align terms with known identities, simplify expressions, or reveal telescoping or convergence properties.

In a finite sum S = sum_{n=a}^{b} f(n), index shifting replaces n with m − k, giving S = sum_{m=a+k}^{b+k} f(m − k). For an infinite series sum_{n=0}^{∞} f(n), shifting to sum_{n=0}^{∞} f(n+1) is valid under appropriate convergence, with attention to the terms that are added or removed by the shift.

Examples help illustrate the idea. Consider S = sum_{n=0}^{4} (n+1). Let m = n+1, so S = sum_{m=1}^{5} m. This shows the same total expressed with a shifted index. Another case is when the series converges: sum_{n=0}^{∞} f(n) = sum_{n=1}^{∞} f(n−1) if the series converges; the shift effectively redistributes terms while preserving the sum, aside from any initial term that is explicitly accounted for.

Applications of indexshifting include proving combinatorial identities (such as those derived from hockey-stick type relations), simplifying sums in generating function calculations, and facilitating telescoping arguments. Common pitfalls include misadjusting the limits after a shift and applying the shift to divergent or conditionally convergent series without proper justification.