# Derivative of sum

1. May 16, 2010

### donutmax

Is the following correct?

$$\frac{d}{dt}\sum_{n=0}^{\infty}\frac{2^{n}t^{n}}{(n+1)!}=\sum_{n=0}^{\infty}\frac{d}{dt}\frac{2^{n}t^{n}}{(n+1)!}$$

2. May 16, 2010

### Char. Limit

Is the derivative of a sum the sum of its derivatives?

Yes, I believe so.

3. May 16, 2010

### flatmaster

Yes. As long as the summation variable is different from the derifination variable.

4. May 16, 2010

### Mute

Usually we (meaning physicists or other applied mathematicians perhaps) don't worry too much about whether or not we can swap a derivative with an infinite sum. It doesn't always work, though, so if you really want to be careful you should check for uniform convergence.

http://en.wikipedia.org/wiki/Uniform_convergence#to_Differentiability

5. May 16, 2010

### Staff: Mentor

derifination???

6. May 17, 2010

### HallsofIvy

As Mute said, we can differentiate (or integrate) an infinite sum "term by term" as long as the convergence is uniform. Fortunately, that is a power series and power series always converge uniformly inside their radius of convergence.

This particular example has infinite radius of convergence so it can be differentiated "term by term" for all x.

7. Apr 22, 2012

### bincy

Dear friends,

What i understood from the previous threads is that if the summation converges summation and differentiation can be interchangeable. Here i assume that summation and differentiation variables are different.

But my doubts are
1. Is it an iff statement?. That is differentiation and summation are interchangeable iff the summation converges.

2.Here my summation is attachment1 . This diverges.
But i suspect that (due to some reasons) attachment2 do not diverge(It would be a function dependant on N after substituting value for x, which is a natural no >=2).

Any sensible suggestions would be really helpful :)

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8. Apr 22, 2012

### HallsofIvy

Do you? Several of the responses told you that is NOT true. I, for example, told you that the differerentiation and summation can be interchanged if the sum converges uniformly. That is a stronger requirement than just saying "converges".

If and only if the sum converges uniformly

Last edited by a moderator: Apr 22, 2012
9. Apr 22, 2012

### Office_Shredder

Staff Emeritus
An example of where differentiation can fail is
$$ln(1-x)=\sum_{n=1}^{\infty} \frac{x^n}{n}$$

At x=-1 this sum converges (to ln(2)) but if we try to differentiate we get
$$\sum_{n=0}^{\infty} x^n$$
and this sum diverges at x=-1, even though the power series is right differentiable at that point

10. Apr 22, 2012

### HallsofIvy

Good example, Office Shredder! As I said before, a power series converges uniformly, and so is differentiable, inside its radius of convergence. Here, -1 is one endpoint of the interval of convergence, not inside it.

11. Apr 22, 2012

### bincy

Thanks.

To Office_Shredder:Your example was really helpful. There a series was converging but its derivative is not. In my case i suspect derivative is converging but series is not.