identify whether the series infinity sigma i=1 8(5/6)^i1 is a convergent or divergent geometric series and find the sum if possible
Question
2 Answer

1. User Answers LammettHash
The sum is convergent. I'll assume the 8 isn't an attempt at using the infinity symbol, so that you have
[tex]\displaystyle\sum_{i=1}^\infty 8\left(\frac56\right)^{i1}[/tex]
This converges because the common ratio between terms is smaller than 1.
The sum is
[tex]\dfrac8{1\frac56}=48[/tex]
since
[tex]\displaystyle\sum_{i=1}^\infty ar^{i1}=a\lim_{n\to\infty}\sum_{i=1}^nr^{i1}=a\lim_{n\to\infty}\frac{1r^n}{1r}=\frac a{1r}[/tex]
if [tex]r<1[/tex]. 
2. User Answers virtuematane
Answer:
The sum of infinite geometric series is:
40
Stepbystep explanation:
We have to find the sum of the geometric series which is given as:
[tex]\sum^{\infty}_{i=1} 8\times (\dfrac{5}{6})^i[/tex]
which could also be written as:
[tex]8\sum^{\infty}_{i=1} (\dfrac{5}{6})^i[/tex]
As we know that any infinite series of the form:
[tex]\sum^{\infty}_{i=1}x^i[/tex]
is convergent if x<1
Here we have:
x=5/6<1
Hence,the infinite series is convergent.
Also we know that for infinite geometric series the sum is given as:
[tex]S=\dfrac{a}{1r}[/tex]
Here we have:
a=5/6 and common ration r=5/6
Hence, the sum of series is:
[tex]8\sum^{\infty}_{i=1} (\dfrac{5}{6})^i=8\times (\dfrac{\dfrac{5}{6}}{1\dfrac{5}{6}})\\\\=8\times (\dfrac{\dfrac{5}{6}}{\dfrac{1}{6}})\\\\=8\times 5\\\\=40[/tex]
Hence, the sum of series is:
40