`sum_(n=0)^oo ((0.3)^n + (0.8)^n)` Find the sum of the convergent series.

`sum_(n=0)^oo((0.3)^n+(0.8)^n)`

`=sum_(n=0)^oo(0.3)^n+sum_(n=0)^oo(0.8)^n`

Now both of the above are geometric series having first term 1 and common ratios `0.3,0.8` respectively.

Geometric series converges to the sum:

`sum_(n=0)^ooar^n=a/(1-r),0<|r|<1`

Using above,

`=1/(1-0.3)+1/(1-0.8)`

`=1/0.7+1/0.2`

`=10/7+10/2`

`=(10*2+10*7)/(14)`

`=90/14=45/7`

The sum of the given convergent series is `45/7`

Comments

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