Property 13
\(a,b,m,n\in\mathbb Z\)
\(\sum_{i=b}^{m}\sum_{k=a}^{n}f(k,i)=\sum_{i=b}^{m}\left(\sum_{k=a}^{n}f(k,i)\right)\)
The outer index is held constant while evaluating the inner sum.
1.
\(\sum_{i=0}^{2}\sum_{k=1}^{3}(k^3+1)=?\)
2.
\(\sum_{i=1}^{3}\sum_{k=0}^{4}(k+2i)=?\)
3.
\(\sum_{k=1}^{10}\sum_{t=1}^{k}\left(t+\samefrac12\right)=?\)
4.
\(\sum_{k=1}^{3}\prod_{n=1}^{k}\samefrac{n}{n+1}=?\)
5.
\(\prod_{k=1}^{3}\sum_{n=1}^{k}(n+1)=?\)
6.
\(\sum_{n=0}^{2}\prod_{i=1}^{2}(i+n)=?\)
7.
\(\prod_{k=-1}^{2}\sum_{r=1}^{3}(r\cdot k)=?\)
Property 14
\(\sum_{k=r}^{n}\samefrac1{k(k+1)}=\sum_{k=r}^{n}\left(\samefrac1k-\samefrac1{k+1}\right)\)
\(\samefrac1{k(k+a)}=\samefrac1a\left(\samefrac1k-\samefrac1{k+a}\right)\)
\(\samefrac1{nk(nk+a)}=\samefrac1a\left(\samefrac1{nk}-\samefrac1{nk+a}\right)\)
1.
\(\sum_{k=1}^{100}\samefrac1{k(k+1)}=?\)
2.
\(\sum_{k=1}^{12}\samefrac1{k(k+2)}=?\)
3.
\(\sum_{k=2}^{7}\samefrac1{k^2-1}=?\)
4.
\(\sum_{k=2}^{9}\samefrac1{k^2-1}=?\)
5.
\(\sum_{k=1}^{10}\samefrac1{k(k+3)}=?\)
6.
\(\sum_{k=1}^{6}\samefrac1{4k^2-1}=?\)
7.
\(\sum_{k=7}^{20}\samefrac1{k(k+1)}=?\)