Question

Consider the following series. n = 1 n The series is equivalent to the sum of two p-series. Find the value of p for each series. P1 = (smaller value) P2 = (larger value) Determine whether the series is convergent or divergent. o convergent o divergent

Answer 1

The given series n^(1/n) is equivalent to the sum of two p-series with P1 = 1/n and P2 = 1. Since P1 is convergent and P2 is divergent, the series is divergent.

Step-by-step explanation:

Series Expansion:

The given series is n1/n. We can rewrite it as (n1)1/n which is equivalent to (n1/n)n = n^(1/n * n) = n. So, the given series is equivalent to the sum of two p-series:

P1 = (smaller value) = 1/n

P2 = (larger value) = 1

Determining Convergence:

For a p-series, if p > 1, the series is convergent. If p <= 1, the series is divergent. In this case, P1 = 1/n, which is less than 1 for all values of n. Therefore, P1 is convergent. P2 = 1, which is greater than 1. Therefore, P2 is divergent. Since P1 is convergent and P2 is divergent, the given series is divergent.