Is the number $$\sum_{n=1}^{\infty}(n \mod 7)10^{-n!}$$ algebraic ? Here, $n \mod 7$ means the natural number is between 0 to 6 which is congruent to n modulo 7.
Justify your explanations, as I don't have any idea what to do with it? It's the first time I have ever seen modulo in series.
DO I need to find the sum which converges to 0.12003 and I don't know how to derive this result. Or is there any other way to find the answer of the question asked.
My questions:
How to do the question given in yellow bar?
Extra credit: How to derive it's convergence values, I know it converges rapidly if we take upto n only 9-10 terms I guess But is there nay more general method ?
The answer is no. The reason for this is something called irrationality measure, which measures how closely a number can be approximated by rationals. See here: http://mathworld.wolfram.com/IrrationalityMeasure.html
There is a theorem that states that non-rational algebraic numbers have an irrationality measure of two (rationals have measure one). But the number you have given has infinite irrationality measure (try proving this once you’ve read the definition!)