`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 the interval.)

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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` .

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial