Σ 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.

SL Core 1.2

k=1∑nc=c+c+⋯+c=n⋅c🚫

SL Core 1.2

k=1∑ncak=ck=1∑nak🚫

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.

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

Exercises