True-False Quiz Question 5: Changing the Coefficients of a Power Series

In this video I show that given that a power series with the powers 6^n is convergent, the same can't be said by a similar power series with negative powers (-6)^n which is actually divergent. First off, we can't use the radius of convergence theorem since if R = 6, we aren't given any information about the end points at x = +/- 6 which can converge or diverge. Instead, we can do a pretty clever trick by rewriting the coefficients such that we obtain an alternating series for the original series, which converges. Doing the same for our given series, we obtain a p = 1 series which is divergent, thus the answer is False.

Time stamps:

• Question 5: 0:00
• Solution: False: 0:18
• Radius of convergence doesn't help us: 4:03
• Modify both series by defining the coefficients: 4:26
• Original series converges by the Alternating Series Test: 5:32
• Changing coefficient for our given series: 9:31
• Our given series is a divergent p = 1 series: 12:06

Full video below:

• Infinite Sequences and Series: Review and True-False Quiz:
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