Some help evaluating series for convergence and divergence

[snowdayb]

Hello everyone i was wondering if you guys could help me with some of my home work problems. i have started most of them and have the work. (wish i could scan it in) anyway i am going to post some problems let me know if u guys can help. i will put the answer i got.

if i am wrong please show the work

1) sum(n=1,inf) 5ⁿ/4ⁿ-3 diverges by ratio test

2) sum(n=1,inf) (-1)ⁿ/n^1/3(cubed root) converges by alt series test

3)sum(n=1,inf) [ln(4n+1)/2n]ⁿ can get this on. do i use alt series test??

4)sum(n=1,inf) cos(2pie-2/n) diverges because of oscilation of cos 2/n term( antother way)

5)sum(n=1,inf) (3n²/6n⁴-5) converges by alt series test

6)sum(n=1,inf) (n+1)!/n³ diverges by ratio test

7)sum(n=1,inf) 1/n(ln n)² clue less

8)sum(n=1,inf) (-1)ⁿ sin²n/n³ ratio test work??

9)sum(n=1,inf) (-1)⁴ n⁴/e⁴ alt series test???

10) power series 1/2n+1 (x+3)ⁿ i worked on this for a while book ans is (-4,-2)


so if anyone could help me out and show their work that would be great. i would like to see the solution to any problem so i can compair it to mine that would be great!!!

 

[tkhunny]

1) sum(n=1,inf) 5ⁿ/4ⁿ-3 diverges by ratio test

I suppose, but can you just observe that the terms fail to decrease? Is "-3" in the denominator? If so, very poor notation.

2) sum(n=1,inf) (-1)ⁿ/n^1/3(cubed root) converges by alt series test

Okay.

3)sum(n=1,inf) [ln(4n+1)/2n]ⁿ can get this on. do i use alt series test??

What alternates about this?

4)sum(n=1,inf) cos(2pie-2/n) diverges because of oscilation of cos 2/n term( antother way)

"pie" is for eating. "pi" is for math and Greek spelling. Clarify the argument. "2pi - 2/n"? "(2pi-2)/n"?

5)sum(n=1,inf) (3n²/6n⁴-5) converges by alt series test

What alternates?

6)sum(n=1,inf) (n+1)!/n³ diverges by ratio test

Good.

7)sum(n=1,inf) 1/n(ln n)² clue less

What did you try? I would have expected at least a Ratio Test.

8)sum(n=1,inf) (-1)ⁿ sin²n/n³ ratio test work??

If you observe that |sin(n)| <= 1, you should see it.

9)sum(n=1,inf) (-1)⁴ n⁴/e⁴ alt series test???

This does not alternate. Think about that exponent on "-1". Also, this CLEARLY diverges unless you wrote it incorrectly. "e" is a constant.

10) power series 1/2n+1 (x+3)ⁿ i worked on this for a while book ans is (-4,-2)

What did the Ratio Test tell you?

You have some good ideas, but a little more care would go a long way.