Bashyboy said:Homework Statement
[itex]\stackrel{lim}{n\rightarrow \infty} (-1)^n \frac{n}{n + 1}[/itex]
Homework Equations
The Attempt at a Solution
The answer is that the limit oscillates between -1 and 1, but I was wondering if there was an analytic was of showing this.
\lim_{n\rightarrow \infty}A sequence is a list of numbers in a specific order. In mathematics, sequences can be finite (with a defined number of terms) or infinite (with an endless number of terms). Each term in a sequence can be represented by a specific rule or pattern.
A limit of a sequence is the value that the sequence approaches as the number of terms increases. It is the value that the terms in the sequence get closer and closer to, but never actually reach.
To solve a limit of a sequence, you can use the formula for the general term (or nth term) of the sequence and plug in increasing values for n. As n gets larger and larger, the terms of the sequence will get closer to the limit value. You can also use algebraic techniques, such as factoring or simplifying, to solve limits of sequences.
This sequence alternates between positive and negative values, with the magnitude of the terms getting closer to 1 as n increases. It is a geometric sequence with a common ratio of -1 and a general term of (-1)^n \frac{n}{n + 1}.
The limit of this sequence is 1, as n approaches infinity. This can be determined by plugging in larger and larger values for n, which will result in the absolute value of the terms getting closer and closer to 1. This can also be proven mathematically using algebraic techniques.