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summations - So,, Ex

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Summations starting from arbitrary numbers: The difference between this type of summations and the actual summations is  The lower limit of the summation does not start from 0 So, to solve such kind of summations, we need to change the lower limit of the  summation to zero from any arbitrary number. Example 1:                                           7 What is the value of  j j=5 This expression can be evaluated in two different ways: First method: We need to change the summation notation so that the lower limit is zero. Note  that this can be accomplished by letting                           m=j-5. Then j= m+5                                                2                             2             2 So the summation will become  (m+5)      =   m + 5                                               m=0                      m=0        m=0 Remember; we have to change the upper limit too. As we are subtracting 5 from 
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