91Ó°ÊÓ

In mathematics, a sequence is a list of numbers written in a definite order. We say that a sequence converges to a limit if, as we proceed along the sequence, the numbers get infinitely close to a particular fixed number. For example, consider the sequence given by \( a_{n} = \sin \left( \frac{\pi}{2} + \frac{1}{n} \right) \). When \( n \to \infty \), the term \( \frac{1}{n} \) becomes very small, trending towards zero.

As a result, \( \frac{\pi}{2} + \frac{1}{n} \) approaches \( \frac{\pi}{2} \). The limit of the sequence then is evaluated by substituting this value into the given function. If the sequence is \( a_{n} \), then \( \lim_{n \to \infty} a_{n} = \sin \left( \lim_{n \to \infty} \left( \frac{\pi}{2} + \frac{1}{n} \right) \right) = \sin \left( \frac{\pi}{2} \right) \). Since \( \sin \left( \frac{\pi}{2} \right) = 1 \), the limit of this sequence is 1.

Thus, the sequence converges to 1. This means that as we increase \( n \), the value of \( a_{n} \) comes increasingly closer to 1 without ever changing after reaching it.

The sine function is an essential trigonometric function with periodic behavior. It oscillates between -1 and 1 as its input varies. The formula for sine is \( \sin(x) \), and it is periodic with a cycle of \( 2\pi \). This means every time you add \( 2\pi \) to the angle, the function repeats its values.

In the context of this sequence, we are interested in small changes around \( \frac{\pi}{2} \). Remember that \( \sin \) is maximized when its argument is \( \frac{\pi}{2} + k(2\pi) \), where \( k \) is an integer, resulting in the maximum value of 1.

The addition of \( \frac{1}{n} \) to \( \frac{\pi}{2} \) introduces a very tiny shift towards \( \frac{\pi}{2} \), given that \( \frac{1}{n} \) becomes negligible for very large \( n \). Hence, the sine function gradually approaches 1 as \( n \to \infty \). This is why the sequence \( \sin \left( \frac{\pi}{2} + \frac{1}{n} \right) \) converges to the value 1.

Continuity is a fundamental property of functions. A function is continuous if small changes in the input result in small changes in the output. For trigonometric functions like sine and cosine, continuity holds true across their entire domain. This means there are no abrupt jumps or gaps in their graphs.

In the scenario of evaluating the sequence \( a_{n} = \sin \left( \frac{\pi}{2} + \frac{1}{n} \right) \), continuity of the sine function allows us to directly apply limits.

Specifically, because \( \lim_{n \to \infty} \left( \frac{\pi}{2} + \frac{1}{n} \right) = \frac{\pi}{2} \), the continuity of the sine function supports that \( \lim_{n \to \infty} \sin \left( \frac{\pi}{2} + \frac{1}{n} \right) = \sin \left( \lim_{n \to \infty} \left( \frac{\pi}{2} + \frac{1}{n} \right) \right) \).

With no sudden changes at \( \frac{\pi}{2} \), this result equals \( \sin \left( \frac{\pi}{2} \right) \), which is clearly 1. Thus, continuity is a key aspect that allows the calculation of limits in sequences involving trigonometric functions like our example above.