Do we always bump the bound up to n=1 when differentiating a power series?

In summary, when differentiating a power series, the starting index is often bumped up by one, and this is typically done when the first term in the series is a constant. This is because when the derivative is taken, the first term will become zero. However, it is ultimately up to personal preference whether to label the starting point as a0 or a1.
  • #1
EV33
196
0
My question is just a concept that I don't understand.

When differentiating a power series that starts at n=0 we bump that bound up to n=1.

My question is do we always do that?

or

Do we only do that when the first term of the power series is a constant and thus when it is differentiated it becomes zero?

My guess is the second case.
 
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  • #2
EV33 said:
My question is just a concept that I don't understand.

When differentiating a power series that starts at n=0 we bump that bound up to n=1.

My question is do we always do that?

or

Do we only do that when the first term of the power series is a constant and thus when it is differentiated it becomes zero?

My guess is the second case.

Huh? n = 0 is just the index. We can call out "starting point" a0 or a1 --- whichever we prefer. And yes, that term will disappear when you take the derivative of ∑anxn.

a0 + a1x + a2x2 + ...

(a0 + a1x + a2x2 + ... )' = a1 + 2a2x + ...

It's as simple as that.
 

1. What is a power series?

A power series is a mathematical series that represents a function as an infinite sum of terms, with each term containing a variable raised to a different power. It is commonly used in calculus to approximate functions.

2. How do you find the radius of convergence for a power series?

The radius of convergence for a power series can be found by using the ratio test, which involves taking the limit of the ratio of consecutive terms in the series. The radius of convergence is the distance from the center of the series to the nearest point where the series diverges.

3. What is the difference between a convergent and a divergent power series?

A convergent power series is one that has a finite limit as the number of terms approaches infinity, while a divergent power series does not have a finite limit and instead goes to infinity or oscillates between different values.

4. How do you differentiate a power series?

To differentiate a power series, you can use the power rule from calculus, which involves multiplying each term by its exponent and then decreasing the exponent by 1. This process is repeated for each term in the series.

5. Can a power series represent any function?

No, a power series can only represent certain types of functions, such as polynomials, exponential functions, and trigonometric functions. Some functions cannot be represented by a power series, such as functions with discontinuities or those that go to infinity at certain points.

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