Respuesta :
Answer:
The sum of given series is 5/3.
Step-by-step explanation:
Given series is
[tex]1+\frac{1}{2}+\frac{1}{10}+\frac{1}{20}+\frac{1}{100}+...[/tex]
We need to find the sum of above series. In this series we alternately multiply by 1/2 and 1/5 to get successive terms.
Isolate the odd and even place terms.
[tex](1+\frac{1}{10}+\frac{1}{100}+...)+(\frac{1}{2}+\frac{1}{20}+...)[/tex]
Now, we have two infinite series.
Sum of an infinite GP is
[tex]S_\infty=\frac{a}{1-r}[/tex]
where, r is first term and r is common ratio, 0 < r < 1.
In [tex]1+\frac{1}{10}+\frac{1}{100}+...[/tex]
First term = 1
Common ratio = 1/10
[tex]S_\infty=\frac{1}{1-\frac{1}{10}}=\frac{10}{9}[/tex]
In [tex]\frac{1}{2}+\frac{1}{20}+...[/tex]
First term = 1/2
Common ratio = 1/10
[tex]S_\infty=\frac{\frac{1}{2}}{1-\frac{1}{10}}=\frac{5}{9}[/tex]
The sum of given series is
[tex]S_\infty=\frac{10}{9}+\frac{5}{9}[/tex]
[tex]S_\infty=\frac{15}{9}[/tex]
[tex]S_\infty=\frac{5}{3}[/tex]
Therefore the sum of given series is 5/3.
