Given the series:
[tex]1-\frac{1}{3}+\frac{1}{9}-...[/tex]
State if the series is convergent or divergent and if convergent, find its sum.
The sequence of terms forms a geometric pattern. Each term is found by multiplying the previous term by a constant number (the common ratio).
Let's find the value of the common ratio by dividing any two successive terms, for example:
[tex]r=\frac{-\frac{1}{3}}{1}=-\frac{1}{3}[/tex]
The common ratio is greater than -1 and less than 1, so the series is convergent.
Now we find the sum of the infinite series by using the formula:
[tex]S=\frac{t_1}{1-r}[/tex]
Where t1 is the first term, that is, t1 = 1.
Substituting:
[tex]S=\frac{1}{1-\left(-\frac{1}{3}\right)}[/tex]
Now we add the numbers to the denominator:
[tex]S=\frac{1}{1+\frac{1}{3}}[/tex]
We have a sum of an integer and a fraction:
[tex]1+\frac{1}{3}[/tex]
To add these numbers, we must get them to have the same denominator, thus:
[tex]1+\frac{1}{3}=\frac{3}{3}+\frac{1}{3}=\frac{4}{3}[/tex]
Operating:
[tex]S=\frac{1}{\frac{4}{3}}[/tex]
To divide by a fraction, we multiply by its reciprocal:
[tex]S=1\cdot\frac{3}{4}=\frac{3}{4}[/tex]
