Question

2. Determine if the following series converge or diverge. Justify your answers, citing any appropriate tests for convergence that you use.

(a) sigma^infinity_n=1 n + 2/(n^5/3 + n + 5 )

(b) sigma^infinity_n=1 (1 − 1/ n)^n ^2

Answer 1

Solution: (a)  

Taking LCM we get:

Applying Limit Comparison Test, where,

where, and will both diverge or converge together. In our case:

Applying Limit Comparison Test (1), we get:

As we can see, the degree of numerator and denominator terms is the same, and therefore, the limit when gets very large will be the ratio of the coefficients of the highest degree terms. In this case, the ratio of the coefficients of terms is 1, therefore, the limit approaches a finite number, that is 1. Now, it can be seen that the series diverges as , and as both and either converge or diverge together when is finite in Limit Comparison Test, we can conclude that the given series is divergent.

(b)

In this problem, we need to apply the Root test in which, for a series , we check the value of . If the limit results in a value greater than 1, then the series diverges. And if the limit results in a value less than 1, then the series converges. Otherwise, if the limit results in exactly 1, then the test is inconclusive. In our case:

Therefore, we need to solve the following limit:

Let this limit be equal to . Therefore:

Put such that as , then . Therefore, the limit becomes:

or,

where,

Taking on both sides, we get:

Applying L'Hopital' Rule and diffentiating numerator and denominator of the limit, we get:

Putting the limit, we get:

Putting in equation (2), we get:

As value of is approximately 2.71828, we get:

which is less than 0. Therefore, the given series converges.