Problem 32 Easy Difficulty

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {1 - n}{2 + 3n} \right) ^n $

Answer

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Video Transcript

since we haven't end power here outside the Prentice is this Suggest that we use the root test. So if we go ahead and call this term and the root test requires that we look at the limit and goes to infinity and through absolute value of Anne So let's go ahead and rewrite this So I'll use the facts from algebra that you've seen before That if you take it and through that's the same thing is raising X to the one over and power So keep the absolute value here This is all to the end power and then are radical becomes one over. And so if you notice here were raising and exponents to another exponents. So we'LL have to use another fact here from from algebra. If you raise an exponents to another exponents, you just multiply the two exponents. So here, when we will supply the end and the one over and they just cancel auto one and that leaves us with the limit as an approaches infinity one minus and over two plus three in an absolute value. Now, to make this easier to deal with, we're letting we can see here that and is bigger than or equal to one just by looking at the summation. So in this case, one minus and absolute value, it's just equal to and minus one because and minus one is always bigger than our vehicles. Zero. So we can rewrite this limit as and minus one over three en plus, too. And if that helps you, you can replace the end with the except this point in use, low Patel's rule to take the limit. In either case, you see that this limit will be won over three. This is less than one, so we conclude that the Siri's convergence and that's by the flutist, and that's our final answer.