Sigma Notation

STEP 2003, Math I, #1: It is given that

can be written in the form

, where , , and are numbers. By setting , , and , obtain four equations that must be satisfied by

, , and and hence show that

Given that

can be written in the form , show similarly that

STEP 2012, Math III, #2: In this question, you may ignore issues of convergence.

(i) Simplify

,

where

is a positive integer, and deduce that

Deduce further that

and hence that

(ii) Show that

Re-writing and Re-indexing