Find the values of $x$ for which the series converges. Find the sum of the series for those values of $x$.
To find: The values of x if the given series is convergent and obtain the sum of the series.
The interval of convergence is (−∞,−2)∪(2,∞) and the sum of the series is xx−2.
The series is ∑n=0∞2nxn.
The geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is converges if |r|<1 and its sum is a1−r, where a is the first term and r is the common ratio of the series.
Obtain the value of x (the interval of converges).
The given series can be expressed as follows,
Clearly, it is geometric series with first term of the series is a=1 and common ratio is r=2x.
Use the Result stated above, the geometric series ∑n=0∞2nxn is converges if |r|<1.
|2x|<1|2||x|<1 [By properties of modulus, |ab|=|a||b|]2|x|<1
Take reciprocal on both sides,
Multiply the above inequality by 2,
|x|2⋅2>1⋅2|x|>2−x>2 (or) x>2
That is, x<−2 (or) x>2.
Thus, the series is converges if x∈(−∞,−2)∪(2,∞).
Therefore, the interval of convergence is (−∞,−2)∪(2,∞).
Obtain the sum of the geometric series.
Since a=1 and r=2x.
Therefore, the sum of the series is xx−2.