Respuesta :
Answer:
The following are the solution to the given points:
Step-by-step explanation:
Given value:
[tex]1) \sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2}\\\\2) \sum ^{\infty}_{k = 1} \frac{1}{(k+6)(k+7)}[/tex]
Solve point 1 that is [tex]\sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2}\\\\[/tex]:
when,
[tex]k= 1 \to s_1 = \frac{1}{1+1} - \frac{1}{1+2}\\\\[/tex]
[tex]= \frac{1}{2} - \frac{1}{3}\\\\[/tex]
[tex]k= 2 \to s_2 = \frac{1}{2+1} - \frac{1}{2+2}\\\\[/tex]
[tex]= \frac{1}{3} - \frac{1}{4}\\\\[/tex]
[tex]k= 3 \to s_3 = \frac{1}{3+1} - \frac{1}{3+2}\\\\[/tex]
[tex]= \frac{1}{4} - \frac{1}{5}\\\\[/tex]
[tex]k= n^ \to s_n = \frac{1}{n+1} - \frac{1}{n+2}\\\\[/tex]
Calculate the sum [tex](S=s_1+s_2+s_3+......+s_n)[/tex]
[tex]S=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+.....\frac{1}{n+1}-\frac{1}{n+2}\\\\[/tex]
[tex]=\frac{1}{2}-\frac{1}{5}+\frac{1}{n+1}-\frac{1}{n+2}\\\\[/tex]
When [tex]s_n \ \ dt_{n \to 0}[/tex]
[tex]=\frac{1}{2}-\frac{1}{5}+\frac{1}{0+1}-\frac{1}{0+2}\\\\=\frac{1}{2}-\frac{1}{5}+\frac{1}{1}-\frac{1}{2}\\\\= 1 -\frac{1}{5}\\\\= \frac{5-1}{5}\\\\= \frac{4}{5}\\\\[/tex]
[tex]\boxed{\text{In point 1:} \sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2} =\frac{4}{5}}[/tex]
In point 2: [tex]\sum ^{\infty}_{k = 1} \frac{1}{(k+6)(k+7)}[/tex]
when,
[tex]k= 1 \to s_1 = \frac{1}{(1+6)(1+7)}\\\\[/tex]
[tex]= \frac{1}{7 \times 8}\\\\= \frac{1}{56}[/tex]
[tex]k= 2 \to s_1 = \frac{1}{(2+6)(2+7)}\\\\[/tex]
[tex]= \frac{1}{8 \times 9}\\\\= \frac{1}{72}[/tex]
[tex]k= 3 \to s_1 = \frac{1}{(3+6)(3+7)}\\\\[/tex]
[tex]= \frac{1}{9 \times 10} \\\\ = \frac{1}{90}\\\\[/tex]
[tex]k= n^ \to s_n = \frac{1}{(n+6)(n+7)}\\\\[/tex]
calculate the sum:[tex]S= s_1+s_2+s_3+s_n\\[/tex]
[tex]S= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{(n+6)(n+7)}\\\\[/tex]
when [tex]s_n \ \ dt_{n \to 0}[/tex]
[tex]S= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{(0+6)(0+7)}\\\\= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{6 \times 7}\\\\= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}+\frac{1}{42}\\\\=\frac{45+35+28+60}{2520}\\\\=\frac{168}{2520}\\\\=0.066[/tex]
[tex]\boxed{\text{In point 2:} \sum ^{\infty}_{k = 1} \frac{1}{(n+6)(n+7)} = 0.066}[/tex]
