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 (−1,5) and the sum of the series is 35−x.
The series is ∑n=0∞(x−2)n3n.
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=x−23.
Use the Result stated above, the geometric series ∑n=0∞(x−2)n3n is converges if |r|<1.
That is, −1<x<5.
Thus, the series is converges if x∈(−1,5).
Therefore, the interval of convergence is (−1,5).
Obtain the sum of the geometric series.
Since a=1 and r=x−23, the sum of the series is computed as follows.
Therefore, the sum of the series is 35−x.