Calculus 2, Lec 23F: The MOST IMPORTANT Ratio Test Example in Calculus??

In Calculus 2, we use the Ratio Test to prove that, for any value of k, the sum ∑k^(n)/n! converges (n goes from 0 to infinity), where n! represents "n factorial": n!=n*(n-1)*...*3*2*1 for n greater than 0, and 0!=1 by definition. This might be the most important infinite series because in Wolfram Mathematica, we see that it converges to e^(k), which many people think is the most important function in Calculus. Sometimes this series is taken to be the definition of e^k (or e^x if you call the variable x), which is called the Taylor series for e^x. This all then leads to the infinite series definition for the number e as e = ∑1/n!).

#calculus2 #RatioTest #apcalculusbc

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