Hint ##\frac{\pi}{n}+1## is irrational (for ##n \in \mathbb N, n\neq 0##).Physicist97 said:Hello! So let's say that you have a sequence ##a{_n}## and the limit as ##n->{\infty}## gives the finite number ##b## not equal to zero. If ##a{_n}## is known to be irrational, and ##a{_n}{_+}{_1}## can be shown to be irrational, does it follow by induction that ##b## is irrational? Is there any theorem that states something equivalent to this, or is this not true for all cases? Thank you!
The rationality of a sequence refers to the pattern or logic behind the numbers in a sequence. It describes how the numbers are related to each other and how they can be predicted or calculated.
A sequence is considered rational if there is a clear and consistent pattern in the numbers, and if the numbers can be expressed as a ratio of two integers. This means that the numbers can be written in the form of a/b, where a and b are whole numbers.
Some common examples of rational sequences include arithmetic sequences (where the difference between consecutive terms is constant), geometric sequences (where the ratio between consecutive terms is constant), and Fibonacci sequences (where each term is the sum of the two previous terms).
Rational sequences can be found in many real-life situations, such as in financial markets, scientific experiments, and natural phenomena. They can be used to make predictions and calculations, and to understand patterns and relationships in data.
No, a sequence can only be either rational or irrational, not both. If a sequence has a clear and consistent pattern, it is considered rational. If there is no discernible pattern or logic behind the numbers, it is considered irrational.