Summation Formulas

    The linearity rule, subtotal rule, and the dominance rule are the basic properties of summations illustrated below. We also give summation formulas for when followed by some examples on taking limits of sums.

Proposition (Basic Rules for Sums) If and are real numbers that do not depend on integers and , then

(i) Constant Term Rule:



(ii) Sum Rule:



(iii)  Scalar Multiple Rule:



(iv)  Linearity Rule:



(v) Subtotal Rule: If then



(vi) Dominance Rule: If for all then

Proposition (Summation Formulas) The summation formulas for when















More summations are listed below.

Example (Summation Formulas) Find the value of the sum

    Solution. We find that

Example (Summation Formulas) Find the value of the sum

    Solution. We find that

Example (Summation Formulas) Find the value of the sum

    Solution. We find that

    It is important to be able to evaluate limits of sums so here are a couple of examples.

Example (Summation Formulas) Evaluate the limit  



    Solution. We find that

Example (Summation Formulas) Evaluate the limit



    Solution. We find that
    

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Summation Formulas
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/summation-formulas.html