Is the Sum of Two Periodic Functions Always Periodic?

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SUMMARY

The sum of two periodic functions, f(x) and g(x), is periodic if the least common multiple (LCM) of their periods, Tf and Tg, exists. Specifically, if Tf/Tg is a rational number, then the sum f(x) + g(x) is periodic. An example discussed is the sum of sin(x) and sin(pi*x), which is not periodic because there is no common x (except zero) that makes both functions zero simultaneously. Therefore, the periodicity of the sum depends on the rational relationship between the periods of the individual functions.

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Cemre
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Hello,

correct me if I am wrong, but as far as I know if 2 functions f(x) and g(x) are periodic with Tf and Tg periods. f(x)+g(x) is also periodic with least common multiple of Tf and Tg.

But; what if that least common multiple doesn't exist?

is "sin(x) + sin(pi*x)" periodic?

there is no x ( except for zero ) which makes both sin(x) and sin(pi*x) zero at the same x.

Regards.
 
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The quotient of the periods, Tf/Tg has to be rational in order for the sum to be periodic.
 

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