The infinite product $$\prod_{n=1}^{\infty}\left(1+10^{-2^n}\right)$$ converges to a specific value, which can be evaluated using advanced techniques in infinite products and series. The discussion highlights the clever solution provided by the user kaliprasad, demonstrating the importance of understanding convergence criteria in mathematical analysis. This evaluation is crucial for mathematicians and students dealing with series and products in their studies.
PREREQUISITESMathematicians, students of advanced calculus, and anyone interested in the evaluation of infinite products and series convergence.
lfdahl said:Evaluate:
$$\prod_{n=1}^{\infty}\left(1+10^{-2^n}\right)$$
kaliprasad said:Llet $x=\prod_{n=1}^{\infty}
(1+10^{-2^n})$
Using $(1-10^{-2^n})(1+10^{-2^n}) = (1+10^{-2^{n+1}})$
We have
$x(1-10^{-2^1})=(1-10^{-2^1})\prod_{n=1}^{\infty}(1+10^{-2^n})$
$=(1-10^{-2^2})\prod_{n=2}^{\infty}(1+10^{-2^n})$
$=\lim{n=\infty}(1+10^{-2^n}) = 1$
or x * .99 = 1 or $x = \frac{1}{.99}=\frac{100}{99}$