### Summation

Sum is one of the basic arithmetic operations. The usual symbol for this operation is ("+") and each term in the sum is called a summand. For example, the sum of \(1\), \(2\) and \(4\) is denoted by

\[1 + 2 + 4.\]

In many situations, the number of summands may be very big making this notation not very practical. In this case, often one hides the intermediate summands with some dots ("..."), making clear how to get back those summands. Thus, the sum of the first thousand natural numbers may be indicated by

\[1 + 2 + 3 + 4 + … + 1000 (*).\]

But some ambiguity may rise when it is not completely clear what the left out summands are. Alternatively, a summation may be represented briefly by the summation symbol (the capital Greek letter sigma)

\[\sum_{i=N_{1}}^{N_{2}}f(i)\]

where \(i\) is the so called index of the summation, that takes integer values between \(N_{1}\) (inferior limit) and \(N_{2}\) (superior limit), and \(f\) is the function that describes the summands. For example, in the summation (*) \(N_{1}=1\), \(N_{2}=1000\) and \(f\) is the identity function:

\[\sum_{i=1}^{1000}i\]

The number of summands of a sum like that is equal to \((N_{2}+1)-N_{1}\). A few more examples:

\[\sum_{i=1}^{7}(2i)=2+4+6+8+10+12+14\]

\[\sum_{j=0}^{4}(2j+1)=1+3+5+7+9\]

\[\sum_{k=-2}^{2}\cos(k\pi)=\cos(-2\pi)+\cos(-\pi)+\cos(0)+\cos(\pi)+\cos(2\pi)\]

\[\sum_{i=-5}^{-2}(ix)=(-5x)+(-4x)+(-3x)+(-2x)\]

With this abbreviated notation, it is also possible to describe sums with an infinite number of summands. For that, it suffices to consider \(N_{1}=-\infty\) and/or \(N_{2}=+\infty\) (the symbol \(\infty\) represents the infinite). For example:

\[\sum_{i=1}^{\infty}\left(\frac{1}{2}\right)^{n}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...\]

\[\sum_{j=-2}^{\infty}j=(-2)+(-1)+0+1+2+3+...\]

\[\sum_{k=-\infty}^{0}(2i+3)=...+(-5)+(-3)+(-1)+1+3\]

\[\sum_{i=-\infty}^{\infty}i^{3}=...+(-2)^{3}+(-1)^{3}+0^{3}+1^{3}+2^{3}+...\]