# Finding interval of convergence

1. Apr 22, 2013

### physics=world

1. Determine a power series, centered at zero for the function ∫f(x)dx. Identify the interval of convergence.

f(x) = ln(x+1) = ∫$\frac{1}{x+1}$

2.

3. i found the power series, which is :

Ʃ ((-1)^(n))(x^(n+1)) / (n+1)

Im okay with that, but i need help on finding the interval of convergence...

to find it i do ratio test.. and i get

-1< x < 1

I need help on checking for the endpoints

the answer in the book is (-1,1]

how do i know that 1 converges? do i just plug in 1 into the function?

2. Apr 22, 2013

### Dick

No, plug 1 into the series. See if the series converges. Do the same thing at x=(-1).

3. Apr 22, 2013

### physics=world

okay so when i plug in 1 into the power series that i found..

i get

Ʃ (-1^n)(1^(n+1)) / (n+1)

how do i know if this converges or diverges?

do i use one of the series test?

4. Apr 22, 2013

### Dick

Sure, use a series test. 1^(n+1)=1. Looks like the alternating series test might be a good one.

5. Apr 22, 2013

### physics=world

hmm when i use the alternating series test

is my a_n = (1^(n+1)) / (n+1)

because when i use that for a_n

i get (1^(infinity)) / (infinity)

or do we ignore (1^(n+1)) because it will always equal to 1?

6. Apr 22, 2013

### Dick

That is NOT the alternating series test. State the alternating series test.

7. Apr 22, 2013

### physics=world

isnt it the lim as n -> infinity of a_n = 0

and

0 < a_n+1 ≤ a_n

8. Apr 22, 2013

### Dick

That's part of the premises. State the rest including the conclusion.

9. Apr 22, 2013

### physics=world

conclusion?

10. Apr 22, 2013

### Dick

The conclusion is that if a series satisfies certain conditions then it converges. Does your series satisfy all of the conditions.

11. Apr 22, 2013

### physics=world

yes the series does satisfy the conditions for 1.

12. Apr 22, 2013

### Dick

Alright. So it converges at x=1. Now try x=(-1).

13. Apr 22, 2013

### physics=world

the lim as n -> infinity of (-1^(n+1)) / (n+1)

im i using the alternate series test correct for this one?

14. Apr 22, 2013

### Dick

No, you left out the (-1)^n factor. Write out the first few terms of the series. Does it really alternate?

15. Apr 22, 2013

### physics=world

okay. so it does not alternate it just stays negative

16. Apr 22, 2013

### Dick

Ok, then you need another test. Does it look like it converges to you?

17. Apr 22, 2013

### physics=world

No it does not look like it will converge

18. Apr 22, 2013

### Dick

If you can give a good reason why then then you are done. It looks pretty similar to the harmonic series 1/n to me.