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`sum_(n=0)^oo (2x)^n` Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of...

Recall the Root test determines the limit as:


`lim_(n-gtoo) root(n)(|a_n|)= L`


or 


`lim_(n-gtoo) |(a_n)|^(1/n)= L`


Then, we follow the conditions:


a) `Llt1` then the series is absolutely convergent


b) `Lgt1` then the series is divergent.


c) `L=1` or does not exist  then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.


For the given series `sum_(n=0)^oo (2x)^n` , we have `a_n = (2x)^n` .


Applying the Root test, we set-up the limit as:


`lim_(n-gtoo) |((2x)^n )^(1/n)| =lim_(n-gtoo) |(2x)^(n*1/n)|`


                                  `=lim_(n-gtoo) |(2x)^(n/n)|`


                                  `=lim_(n-gtoo) |(2x)^1|`


                                  `=lim_(n-gtoo) |(2x)|`


                                  ` =|2x|`


Applying  `Llt1` as the condition for an absolutely convergent series, we let `L=|2x|` and set-up the interval of convergence as:


`|2x|lt1`


`-1 lt2xlt1`


Divide each part by 2:


`(-1)/2 lt(2x)/2lt1/2`


`-1/2ltxlt1/2`


The series may converges when `L =1` or `|2x|=1` . To check on this, we test for convergence at the endpoints: `x=-1/2` and `x=1/2` by using geometric series test.


The convergence test for the geometric series `sum_(n=0)^oo a*r^n`  follows the conditions:


a) If `|r|lt1`  or `-1 ltrlt 1` then the geometric series converges to `a/(1-r)` .


b) If `|r|gt=1` then the geometric series diverges.


When we let `x=-1/2` on `sum_(n=0)^oo (2*(1/2))^n ` , we get a series:


` sum_(n=0)^oo 1*(-2/2)^n =sum_(n=0)^oo 1*(-1)^n`



It shows that `r=-1` and `|r|= |-1|=1` which satisfies |r|>=1. The series diverges at the left endpoint.


When we let `x=1/2` on `sum_(n=0)^oo (2*1/2)^n` , we get a series:


`sum_(n=0)^oo 1*(2/2)^n =sum_(n=0)^oo 1*(1)^n`


It shows that `r=1` and `|r|= |-1|=1` which satisfies `|r|gt=1` . The series diverges at the right endpoint.


Conclusion:


The interval of convergence of the power series `sum_(n=0)^oo (2x)^n ` is `-1/2ltxlt1/2` .

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