And what's interestingĪbout this is we've proven to ourselves, in the videosĪbout geometric series, that if the common ratio, If it is not review, I encourage you to watch the videos on geometric series. When you go from one term to another you are just multiplying, youĪre just multiplying by R, and this is all review. To R, or R to the first power, and you see that here. That same thing with the N, that's just going to be equal Well, this is just going to be equal to R. Which is the ratio between consecutive terms, is going There are a few things we'veĪlready thought about here. Plus R to the K plus one plus R to the K plus two and keep going on and on and on forever. Infinity of R to the N, which would be R to the K Infinite geometric series starting at N equals K to We already have a lot of experience with the geometric series. Let me know if that didn't fully help and I can try explaining differently If you'd like to see it on a smaller scale, try (100x^2 + 100x)/(x^3) it starts with the numerator being larger, but eventually the denominator is bigger. if the denominator keeps getting bigger than the numerator than eventually it will equal 0. What does that mean? well, a large denominator makes the fraction get closer and closer to 0. the numerator may start out bigger, but as you head toward infinity, the larger exponent will always make a bigger term, so the denominator will get bigger than the numerator. Nowwhat happens as n gets bigger and bigger? n^11 will always be bigger than an^9 + bn^8 +. the n^10 in the numerator and denominator kinda cancel out, or at least they will be the same number no matter what n is, so we only need to worry about the rest. You could maybe look at it as n^10 + n^11. + z where a through z are some real numbers. So, the numerator is a massive polynomial, but the largest term is n^10, so it will be n^10 + an^9 +bn^8 +.
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