# True-False Quiz Question 4: Radius of Convergence Theorem and Comparison Test

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• Published on Jan 02, 2024

In this video I show that if we are given that a power series of the form 6^n is convergent, then a power series of the form (-2)^n is also convergent by the radius of convergence theorem and also the comparison test. The radius of convergence theorem states that a power series has only 3 possibilities for convergence: the series converges only for x = a = 0 in our case, the series converges for all x, or it converges for |x - a| is less than R. In our case the 6^n means that it is convergent for R is less than 6, which is greater than 2, thus our given series is also convergent. Likewise, we can compare the positive form of our power series with the convergent power series to show that it too is convergent, since all the terms would be less than the convergent series. And if a series is absolutely convergent than it is also convergent, thus proving our case.

Timestamps:

• Question 4: 0:00
• Solution 1: True: Radius of Convergence Theorem: 0:18
• Solution 2: Comparison Test: 7:02

Full video below:

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