Question about the derivative of this sum and where n starts

In summary, the conversation discusses differentiating the series 1/(1-x) and its derived series 1/(1-x)^2. The original series starts at n=0 while the derived series starts at n=1. The reason for this is that starting at n=0 could cause issues with x=0, so it is recommended to start at n=1 for the derived series. However, both series technically start with 1, as plugging in n=0 for the original series will result in 1.
  • #1
Frankenstein19
56
0
Ok so when differentiating

1/(1-x)= Σ xn from n=0 to infinity

the book says it is 1/(1-x)^2 = Σ n*(x)n-1 from n=1 to infinity

i don't understand why the original sum starts at 0 and then the derived sum starts at 1
 
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  • #2
Both start with the number 1. If you like you may start the second series at 0, too. It doesn't make a difference because ##0 \cdot x^{-1} = 0.## However, in this case one unnecessarily gets into trouble with ##x=0.##
 
  • #3
Expand the sum and take the derivative and then rewrite the result in sum notation.
 
  • #4
both starts with 1 because when you put ##n=0## in first series,the answer will be ##x^0=1##
 

FAQ: Question about the derivative of this sum and where n starts

1. What is the derivative of a sum?

The derivative of a sum is equal to the sum of the derivatives of each individual term. This is known as the sum rule of derivatives.

2. How do you find the derivative of a sum when n starts at a specific value?

To find the derivative of a sum where n starts at a specific value, you would simply apply the sum rule of derivatives to the terms with n starting at that value, while treating the other terms as constants.

3. Is the derivative of a sum the same as the sum of the derivatives?

Yes, the derivative of a sum is the same as the sum of the derivatives of each individual term. This is a fundamental property of derivatives known as the linearity property.

4. Can the derivative of a sum be simplified?

Yes, the derivative of a sum can often be simplified by using algebraic techniques such as factoring or the distributive property. However, the simplified form may not always be possible or desirable, depending on the specific problem.

5. Are there any special cases when finding the derivative of a sum?

Yes, there are some special cases when finding the derivative of a sum. For example, if the sum contains a constant term, its derivative would be 0. Additionally, if the sum contains a constant multiple of a variable, the derivative would be equal to that constant multiple.

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