Question

What does it mean to say that derivatives commute with sums? Do that limits commute with sums? Does that sum written in sigma notation commute with sums? Express your answers in words and algebraically.

Short answer

Ans: The summation of the sum of the functions is the same as that of the individual sum of the summation of that functions.

$\sum \left(f\right(x)+g(x\left)\right)=\sum f\left(x\right)+\sum g\left(x\right)$

Step-by-step solution

Step 1. Given information.

given,

the sum is written in sigma notation commute with sums

Step 2. Derivatives commute with sums:

The derivative of the sum of the functions is the same as that of the individual sum of the derivative of that function.

$\frac{d}{dx}\left(f\right(x)+g(x\left)\right)=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$

Step 3. Limits commute with sums:

The limit of the sum of the functions is the same as that of the individual sum of the limits of that functions.

$\underset{x\to a}{lim} \left(f\right(x)+g(x\left)\right)=\underset{x\to a}{lim} f\left(x\right)+\underset{x\to a}{lim} g\left(x\right)$

Step 4. Sums are written in sigma notation commute with sums:

The summation of the sum of the functions is the same as that of the individual sum of the summation of that functions.

$\sum \left(f\right(x)+g(x\left)\right)=\sum f\left(x\right)+\sum g\left(x\right)$