NATURAL FIBONACCI SEQUENCE SERIES

However, hopefully that's understandable I'm not sure of your mathematical background.

NATURAL FIBONACCI SEQUENCE SERIES

The article I'm adapting this from is written a little better, so you can see what is going on, but they manipulate the infinite series in this way, which is not technically permitted. So we get that the partial sums converge to 1/89 and so x = 1/89, as we expected. S( N) = 1/89 - 89/100*.įinally, as N gets very large, those last three terms get very close to 0, so when we consider the infinite series, they do not contribute at all. Since fib(0) = fib(1) = 1, we can actually get rid of the second and third terms as well. So everything cancels out, except for the first and last few terms (this is a so-called telescoping sum).Īfter this nice cancellation, we're left with: S( N)-1/10*S( N)-1/100*S( N) = fib(0)/10 2 + fib(1)/10 3 - fib(0)/10 3 - fib( N)/10 N+3 - fib( N-1)/10 N+2 - fib( N)/10 N+4. But this is just 0, since fib(2) = fib(1) + fib(0). For example, we get fib(2)/10 4 - fib(1)/10 4 - fib(0)/10 4 from each term, respectively. Now look at the terms that have the same power of 10 in them. Notice that I've distributed the 1/10 and 1/100 into the sums by increasing the exponent of the 10s. However, we're going to use the fact that fib( n) = fib( n-1) + fib( n-2) to our advantage. + fib( N)/10 N+2.Ĭonsider S( N)-1/10*S( N)-1/100*S( N), which is something not immediately obvious that we should look at. This will allow us to treat it like a regular finite sum, as we're used to, and then use limits to make N arbitrarily large, i.e. we're going to stop N terms into the series. It's actually important to prove that this series converges, but I'm not going to do that. So, let's call the number that we're interested in x, i.e. Let fib( n) be the nth Fibonnaci number, i.e. Okay, so I'm adapting this from here but modifying it to be a bit more rigorous / to my liking. We make our world significant by the courage of our questions and by the depth of our answers.

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Natural fibonacci sequence