Problem: Investigate convergence of the following power series:
Solution: Is this really a power series? Actually, it is, we can rewrite it as
We see that the center of this power series is
We used notation Ak for terms of this power series,
since "ak" (that you usually find in the Root test) is
used for coefficients in the context of power series. In the denominator we
used the fact that the
We know that the series converges absolutely if rho is less than one, thus the condition is
The radius of convergence is
The interval of convergence is therefore at least
(−2 − 1/2,−2 + 1/2) = (−5/2,−3/2).
What is unclear is the status of endpoints, we need to check on them individually.
Endpoints:
x = −5/2: We substitute into the given series and
obtain the series
What test do we use? Since the terms are not positive, we cannot use the usual tests. The natural choice here is the Alternating series test with
The sequence
x = 4: We substitute into the given series and obtain the series
What test do we use? This series is a prime candidate for some comparison. Would plain comparison test do? We can naturally estimate
and the series on the right is convergent (see the p-test). Thus by the comparison test, the series we investigate converges.
Thus both endpoints belong to the region of convergence, the above calculations show that we also have absolute convergence there.
Conclusion: The given series has the center