Σ summation notation
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Summation notation, properties of sums
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Understanding summation notation
SL Core 1.2
As a shortcut for writing out long sums, we can use the symbol ∑ with the following "settings":
n=(start value)∑end value of n(term depending on n)
Here n is called the index, but other letters can also be used in place of n.
Sum of a constant
SL Core 1.2
k=1∑nc=c+c+⋯+c=n⋅c🚫
Sums with scalar multiples
SL Core 1.2
k=1∑ncak=ck=1∑nak🚫
Sum of a sum
Essentially, any ∑ of a sum can be broken into two ∑ 's.
k=1∑n(ak+bk) =(a1+b1)+(a2+b2)+⋯+(an+bn)=a1+a2+⋯an+b1+b2+bn=k=1∑nak+k=1∑nbk
Or vice versa! If two sums have the same start and stop index (eg k=1 up to n), they can be merged.
Splitting a sum in Σ form
SL Core 1.2
For any series of the form k=1∑nak and any integer m between 1 and n, we can split the series at the index m:
k=1∑nak=k=1∑mak+k=m+1∑nak
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