Worked Examples
Write down the values of (−1)^0,\ (−1)^1,\ (−1)^2,\ (−1)^3 etc.
Generalise your answers.
Show me1, −1, 1, −1; clearly the sequence continues to alternate in this way.
Write down the first three terms and the last term of.
\sum_{r=0}^{10}\left(-1\right)^r\frac{1}{2^r}
Show me(−1)^0 \frac{1}{{2^0}} + (−1)^1 \frac{1}{{2^1}} + (−1)^2 \frac{1}{{2^2}} = 1 - \frac{1}{2} + \frac{1}{4}
and last term is (−1)^{10} \frac{1}{{2^{10}}} = \frac{1}{{2^{10}}} \left(= \frac{1}{1024}\right)
\sum_{r=0}^{10}\left(-1\right)^{r+1}\left(\frac{r^2}{r^2+1}\right)
Show me\left(-1\right)^1\frac{0^2}{\left(0^2+1\right)}+\left(-1\right)^2\left(\frac{1^2}{1^2+1}\right)+\left(-1^3\right)\left(\frac{2^2}{2^2+1}\right)=-\frac{1}{2}+\frac{1}{2}-\frac{4}{5}
and last term is (−1)^{11} \left(\frac{10^2}{10^2 + 1}\right)= -\frac{100}{101}
How can you write the series
100 − 100 × (0.8) + 100 × (0.8)2 ... to n terms using Σ notation?
Show me\sum_{r=0}^n100\left(-8\right)^r or \sum_{r=0}^n\left(-1\right)^r\ 100\times\left(0.8\right)^r
It is also useful to know these three important results, the first of which is the sum of the arithmetic progression,
1, 2, 3, ... , n
\sum_{r=1}^nr=\frac{1}{2}n\left(n+1\right)
\sum_{r=1}^nr^2=\frac{1}{6}n\left(n+1\right)\left(2n+1\right)
\sum_{r=1}^nr^3=\frac{1}{4}n^2\left(n+1\right)^2
The useful fact that \sum_{\ }^{\ }r^3 = (\sum_{\ }^{\ }r)^2 is a coincidence (if there is such a thing in maths). It is not possible to extend this to find \sum_{\ }^{\ }r^4, \sum_{\ }^{\ }r^5 etc. Formulae do exist for sums of higher powers, but they are somewhat cumbersome and seldom useful.
Exercises
