Find the sum of each of the following infinite series. If a series does not have a sum, write no sum as the answer. 1/2+1+2+4... 9-3+1-1/3+1/9...

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First series is geometric series with ratio `r=2` and geometric series is convergent only if `|r|<1`, hence the first series isn't convergent. **In other words sum of the first series is** `oo.`

The other series we can look as the sum of two geometric series. One with general term `a_n=(1/9)^(n-2)` and other with general term `b_n=-(1/3)^(n-2),` `n=1,2,3,ldots`

Hence sum of your series is

`S=sum_(n=1)^ooa_n+sum_(n=1)^oob_n` **(1)**

Sum of infinite geometric series is calculated by formula

`sum_(n=1)^ooa_1r^n=(a_1)/(1-r)`` `

Hence we have

`sum_(n=1)^ooa_n=9/(1-1/9)=9/(8/9)=81/8`

`sum_(n=1)^oob_n=-3/(1-1/3)=-3/(2/3)=-9/2`

Now we put that into (1) to get

`S=81/8-9/2=(81-36)/8=45/8` **<-- Your solution**

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