\(\sin(x)\) and \(\cos(x)\) are harmonic functions.Their series can be given as,
sinxcosx==x−3!x3+5!x5−7!x7+⋯=n=0∑∞(2n+1)!(−1)nx2n+11−2!x2+4!x4−6!x6+⋯=n=0∑∞(2n)!(−1)nx2n
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+2!x2−3!x3+4!x4−5!x5+...
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?
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Top NewestThe problem is that sinx and cosx 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.
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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.
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Of course it does change the property, but how can you say that it's bounded just by looking at the signs?
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