Infinite Series Convergence and which Convergence tests can be used for part (a) and part (b).

Question

(a) Check all of the following that are true for the series ∞∑n=1[(6^n/(4^n−1))]

A. This series converges 

B. This series diverges 

C. The integral test can be used to determine convergence of this series. 

D. The comparison test can be used to determine convergence of this series. 

E. The limit comparison test can be used to determine convergence of this series. 

F. The ratio test can be used to determine convergence of this series. 

G. The alternating series test can be used to determine convergence of this series.

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(b) Check all of the following that are true for the series ∞∑n=1[((n^4−6)cos⁡(nπ))/n^5] 

A. This series converges 

B. This series diverges 

C. The integral test can be used to determine convergence of this series. 

D. The comparison test can be used to determine convergence of this series. 

E. The limit comparison test can be used to determine convergence of this series. 

F. The ratio test can be used to determine convergence of this series. 

G. The alternating series test can be used to determine convergence of this series.

Answer 1

Let's start with (a):

A. This series converges 

B. This series diverges.

1. We know this by using any of the tests below, or by using the nth Term Test (a much easier approach than the other tests).

C. The integral test can be used to determine convergence of this series. 

1. The integral test is useful when a term and the derivative of that term appear in the problem, making the integral easy to perform. We do not see that in this case -- the integral is not easy to perform. We also notice that the integral test requires the values of our function to be decreasing, but the values of the function corresponding to this series would be increasing and so the integral test does not meet its conditions and cannot be used.

D. The comparison test can be used to determine convergence of this series. 

1. Our series looks a lot like the series of (6/4)^n = (3/2)^n -- a divergent geometric series.

E. The limit comparison test can be used to determine convergence of this series. 

1. Compare using the same series as above -- (6/4)^n

F. The ratio test can be used to determine convergence of this series. 

1. The ratio test is useful when there are factorials and/or exponentials in our series. In this series, 6^n and 4^n are both exponentials.

G. The alternating series test can be used to determine convergence of this series.

1. This series is not an alternating series, as there is no alternating factor, and so the alternating series test should not be used.

Now, for (b), you should use the same logic as laid out above, noticing that the cos⁡(nπ) term is an alternating factor.