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 series.
The interval of convergence is (−3,−1) and the sum of the series is −x+2x+1.
The series is ∑n=1∞(x+2)n.
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=x+2 and common ratio is r=x+2.
Use the Result stated above, the geometric series ∑n=1∞(x+2)n is converges if |r|<1.
Thus, the series is converges when x∈(−3,−1).
Therefore, the interval of convergence is (−3,−1).
Obtain the sum of the geometric series.
Since a=x+2 and r=x+2, the sum of the series is computed as follows.
Therefore, the sum of the series is −x+2x+1.