Answer:
1. Diverges
2. Converges
3. Diverges
Step-by-step explanation:
Solution:-
Limit comparison test:
- Given, ∑ and suppose ∑ such that both series are positive for all values of ( n ). Then the following three conditions are applicable for the limit:
Lim ( n-> ∞ ) = c
Where,
1) If c is finite: 0 < c < 1, then both series ∑ and ∑ either converges or diverges.
2) If c = 0, then ∑ converges only if ∑ converges.
3) If c = ∞ or undefined, then ∑ diverges only if ∑ diverges.
a) The given series ∑ is:
(n = 1) ∑^∞
- We will make an educated guess on the comparative series ∑ by the following procedure.
(n = 1) ∑^∞
- Apply the limit ( n - > ∞ ):
(n = 1) ∑^∞ .... The comparative series ( ∑ )
- Both series ∑ and ∑ are positive series. You can check by plugging various real number for ( n ) in both series.
- Compute the limit:
Lim ( n-> ∞ )
Lim ( n-> ∞ )
- Apply the limit ( n - > ∞ ):
Lim ( n-> ∞ ) = = 1 ... Finite
- So from first condition both series either converge or diverge.
- We check for ∑ convergence or divergence.
- The ∑ = ( 1 / 2n ) resembles harmonic series ∑ ( 1 / n ) which diverges by p-series test ∑ ( ) where p = 1 ≤ 1. Hence, ∑
- In combination of limit test and the divergence of ∑, the series ∑ given also diverges.
Answer: Diverges
b)
The given series ∑ is:
(n = 1) ∑^∞
- We will make an educated guess on the comparative series ∑ by the following procedure.
(n = 1) ∑^∞
- Apply the limit ( n - > ∞ ) in the denominator for ( 5 / n ), only the dominant term n^(3/2) is left:
(n = 1) ∑^∞ .... The comparative series ( ∑ )
- Both series ∑ and ∑ are positive series. You can check by plugging various real number for ( n ) in both series.
- Compute the limit:
Lim ( n-> ∞ )
Lim ( n-> ∞ )
- Apply the limit ( n - > ∞ ):
Lim ( n-> ∞ ) = = 1 ... Finite
- So from first condition both series either converge or diverge.
- We check for ∑ convergence or divergence.
- The ∑ = ( ) converges by p-series test ∑ ( ) where p = 3/2 > 1. Hence, ∑
- In combination of limit test and the divergence of ∑, the series ∑ given also converges.
Answer: converges
Comparison Test:-
- Given, ∑ and suppose ∑ such that both series are positive for all values of ( n ).
-Then the following conditions are applied:
1 ) If ( - ) < 0 , then ∑ diverges only if ∑ diverges
2 ) If ( - ) ≤ 0 , then ∑ converges only if ∑ converges
c) The given series ∑ is:
(n = 1) ∑^∞
- We will make an educated guess on the comparative series ∑ by the following procedure.
(n = 1) ∑^∞
- Apply the limit ( n - > ∞ ) in the numerator for ( 4 / 3^n ), only the dominant terms ( 3^n ) and ( 2^n ) are left:
(n = 1) ∑^∞ ... The comparative series ( ∑ )
- Compute the difference between sequences ( - ):
, for all values of ( n )
- Check for divergence of the comparative series ( ∑ ), using divergence test:
∑ = (n = 1) ∑^∞ diverges
- The first condition is applied when ( - ) ≥ 0, then ∑diverges only if ∑ diverges.
Answer: Diverges