# Is there a summation rule for continuous functions?

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Is there a summation rule for continuous functions? i.e.: does #sum_(n=1)^"∞" f(x)# = #f(sum_(n=1)^"∞" x)# ?

Obviously, I know that isn't true, but is there some other transformation which applies to all continuous functions, or is there not?

If there isn't a rule for *all* functions f(x), is there any for specifically any of the three primary trig functions?

Is there a summation rule for continuous functions? i.e.: does

Obviously, I know that isn't true, but is there some other transformation which applies to all continuous functions, or is there not?

If there isn't a rule for *all* functions f(x), is there any for specifically any of the three primary trig functions?

##### 1 Answer

The sum of two continuous functions is continuous.

Let

and clearly the result extends to the sum of any finite sum of continuous functions.

However let

For the values of

that is the sum of the series. But while every partial sum is defined and continuous in

For instance consider the functions:

that are defined and continuous for every

that we can find based on the sum of the geometric series is not defined for

Similarly we can demonstrate based on Fourier analysis that the series:

is defined for every

A sufficient condition for the sum of the series to be continuous, is that the series is **totally** convergent, that is for every

and that the series:

is convergent.