sum rules What are the most important rules of double sums Below are some rules I encountered are they all correct and complete Offerings of clear intuition or proofs or other additions are most welc
26 When we deal with summation notation there are some useful computational shortcuts e g sum limits i 1 n 2 3i sum limits i 1 n 2 sum limits i 1 n 3i 2n sum limits i 1 n 3i However I don t think I know all the useful shortcuts here Are there other computational tricks one should be aware of By itself the first term already gives 21 0819 the sum of first and second term gives approx 22 6629
sum rules
sum rules
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Prepare to combine 1 and 2 by introducing N max N1 N2 N max N 1 N 2 and C max C1 C2 C max C 1 C 2 Add 1 and 2 3 T1 n T2 n C1f n Check that for any two real numbers a b we have a b 2 max a b Use 4 in 3 to obtain T1 n T2 n 2C max f n g n when n N Conclude that 0 The sum rule for derivatives states that the derivative of a sum is equal to the sum of the derivatives In symbols this means that for f x g x h x we can express the derivative of f x f x as f x g x h x For an example consider a cubic function f x Ax 3 Bx 2 Cx D Note that A B C and D are all constants Now we will make use of three other basic
The sum rule would tell you how many ways to pick either a woman or a man who is not her husband that doesn t really make sense at all how can you only pick a man who is not her husband so the sum rule cannot be applicable here I have sum k n P k x k Am I allowed to split it up into two sums so I have it like sum k n P k sum k nx k
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Similarly in a sum to product conversion the plausible absolute value for the constant is 2 With these we can figure out everything in the formula except for what s inside the arguments of cos sin and whether to add subtract and also that front minus sign in the formula for cos a cos b If you are given a nested sum b j a d j k c j zj k each individual summand zk j depends on the outer variable j as well as on the inner variable k Sometimes a factor depending only on the outer variable j can be pulled out but forget this at the moment The essential point is to get a clear picture of the set S
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