Youtor
∑2nn31 What Happens If We Use BOTH Divergence Test AND Ratio Test
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Here's an infinite series example in Calculus: does ∑2^n/(n^3+1) converge or diverge? At first, it looks like the terms decrease so that it might converge. But, in fact, the exponential function 2^n goes to infinity much faster than the cubic function n^3+1, so the terms do not decrease to zero and the series diverges by the Divergence Test. If we also apply the Ratio Test to this series, we also conclude divergence because the ratio |a_{n+1}|/|a_{n}| converges to 2, which is greater than 1. In fact, any time the ratio in the Ratio Test converges to something greater than 1, the (simpler) Divergence Test could have been used instead. With regard to p-series ∑1/n^p, the Ratio Test is not helpful because the limit of |a_{n+1}|/|a_{n}| is always equal to 1. • Calculus 2, Lecture 24A • #calculus2 #SeriesTest #RatioTest • Links and resources • =============================== • 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... • 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ • 🔴 Follow me on Twitter: / billkinneymath • 🔴 Follow me on Instagram: / billkinneymath • 🔴 You can support me by buying Infinite Powers, How Calculus Reveals the Secrets of the Universe , by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. • 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ • 🔴 Desiring God website: https://www.desiringgod.org/ • • AMAZON ASSOCIATE • As an Amazon Associate I earn from qualifying purchases.
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