Little Help..Calculus 3?

Question

Determine Convergence or Divergence

Sum from 1 to inf of e^(1/n) / n
-What test would I use?

Sum from 0 to inf of (1+sin n) / 10^n
- What would I use?
- Would I use root test and put the whole thing to the nth power?

Sum from 0 to inf ((4n+3) / (2n-1))^n
- I tried root test but am stuck

Sum from 1 to inf (-1^n*3^n) / (n*2^n)
-I got lim n->inf 3n / (2(n+1))..how do I tell if that converges or diverges?

Answer 1

1) try the integral test.

2) comparison test.
| (1+sin n) / 10^n | < 2 / n² , and Σ 2 / n² converges,
then Σ (1+sin n) / 10^n converges.

3) as n becomes large, the terms approach (4n/2n)^n, or simplifying, they approach 2^n, which clearly does not converge since the terms do not approach zero in the limit.

4) try ratio test or alternating series test.

Answer 2

1) comparison test with 1/n
e^(1/n) is always greater than 1 so e^(1/n)/n is greater than 1/n. Since Sum 1/n diverges, then Sum e^(1/n)/n divergers
2) you have the summation of two sereis:1/(10^n) and sin (n)/ (10^n). Both of them converge, so their summation coverges too.
3) take the limit of the division when n approches infinity
it would be 2. Since 2 is greater than 1 then it diverges
4)take the limit when n approches infinity (ignore -1^n becuase it's only means the sign chnages)
it diverges becuase the limit is greater than 1
your limit is not correct. use L'hospital role

Answer 3

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