Question

For each of the following series, tell whether or not you can apply the 3-condition test (i.e. the alternating series test). Enter D if the series diverges by this test, C if the series converges by this test, and N if you cannot apply this test (even if you know how the series behaves by some other test).1. ∑n=2 to [infinity]((−1)^n(n^4+2^n))/(n^3−1)2. ∑n=1 to [infinity](−1)^n/n^53. ∑n=1 to [infinity] ((−1)^ncos(n))/(n^2)4. ∑n=1 to [infinity] (((−1)^n(n^3+1))/(n^3+7)5. ∑n=1 to [infinity]((−1)^n(n^10+1))/e^n6. ∑n=1 to [infinity](((−1)^n(n^3+1))/(n^4+1)

Answer 1

Answer:

1. D  2. C  3. C  4. N 5. C  6. N

Step-by-step explanation:

The Alternating Series Test

∑(-1)nBn converges when the following two conditions are met:

(i) lim Bn = 0 and (ii) {Bn} is (eventually) decreasing.

Note: The Alternating Series Test doesn't apply when either of the conditions is not met, and so never is a test for divergence.  If the first condition isn't met, then the "n-th term test" will show divergence.

1. Bn = (n^4+2^n))/(n^3−1)

lim (n^4+2^n))/(n^3−1) = 0 but (ii) the function is increasing as n increases, Hence diverges

2. Bn = 1/n^5 lim Bn = 0. this is a converging p series as P>5 satisfies the condition for P series convergence P > 1 converges, 0<P<=1 diverges and (ii) is satisfied. Hence Converges

3. Bn = cos(n))/(n^2) ; cos(n) has no limit since it dangles between 0 and 1 and limit is unique. leaving 1/n^2 to converge according to the condition for P series convergence P > 1 converges, 0<P<=1 diverges and (ii) is satisfied. Hence Converges.

4. Bn = (n^3+1))/(n^3+7) ; lim Bn as n tends to infinity = 0. (ii) can be confirmed by limit compression test

5. Bn = (n^10+1))/e^n; lim Bn = 0 also Bn is a decreasing function since the denominator is an increasing function and (ii) is satisfied. Hence Converges.

6. Bn =(n^3+1))/(n^4+1) ; lim Bn as n tends to infinity = 0. (ii) can be confirmed by limit compression test