Harmonics

$\sin(x)$ and $\cos(x)$ are harmonic functions.Their series can be given as,

$\begin{aligned} \sin x &=& x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots = \sum_{n=0}^\infty \dfrac{(-1)^n x^{2n+1}}{(2n+1)!} \\ \cos x &=& 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots = \sum_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!} \end{aligned}$

Here, we can see that both the above series have maximum magnitude -: $1$ . The speciality of both the functions is that there is $+$ and $-$ signs alternatively. Thats why for any big or small values of $x$ , both the above series converges to $1$ in magnitude.

Now, if we look at the expansion of $e^{(-x)}$ ,

$e^{(-x)} = 1-x+\dfrac{x^2}{2!}-\dfrac{x^3}{3!}+\dfrac{x^4}{4!}-\dfrac{x^5}{5!}+...$

Here also the same situation occurs. But why it is not a function like $sine$ and $cos$ ? Why its value is not repeated like $sine$ and $cos$ ?

#Geometry

Easy Math Editor

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Comments

The problem is that $\sin x$ and $\cos x$ do not converge, they are just bounded between $[-1,1]$ .
Proving this (and periodicity) is not direct from the series expansions, irrespective of the $+$ and $-$ signs.
$e^{-x}$ is still a function, however.

Ameya Daigavane - 5 years ago

Thank you, Sir. I think the power changes the property. As in $sin$ and $cos$ the powers are odd and even, so even for any higher value of $x$ , due to + and -, it will be bounded to $-1$ to $1$ .

Akash Shukla - 5 years ago

Of course it does change the property, but how can you say that it's bounded just by looking at the signs?

Ameya Daigavane - 5 years ago

@Ameya Daigavane – If + and - are not alternate or if + or - sign are repeated after two terms then I think it will not repeat its value.

Akash Shukla - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$