Question

To find the terms an and the sum of the series ∑n=1[infinity]an, you can use the given information that the nth partial sum sn is:

sn=n+3n−3
We can start by finding an expression for an in terms of sn. Then, we'll find the value of a1 and the sum of the series

Answer 1

To find the nth term (a_n) of the series given the nth partial sum s_n, we use the relation a_n = s_n - s_(n-1). Applying this, a_n simplifies to a constant value, 3. Since the series is infinite and the terms are constant, the sum of the series is divergent and not finite.

Step-by-step explanation:

The student's question involves finding the nth term (an) and the sum of the series ∑n=1[infinity]an, using the given nth partial sum sn = n + 3n - 3. To determine an, we consider the relationship between consecutive partial sums: an = sn - sn-1. So a1 is the first term of the series.

For s1, which is the sum of the first term of the series, we have s1 = 1 + 3(1) - 3 = 1, thus a1 = 1. We apply the relation an = sn - sn-1 to obtain an = (n + 3n - 3) - [(n-1) + 3(n-1) - 3 ] = n + 3n - 3 - n + 1 - 3(n-1) which simplifies to an = 3.

If the terms of the series are constant (each an is the same), then the series is a multiple of a sum of ones. Since the series is infinite, the sum of the series diverges to infinity. Therefore, there is no finite sum for ∑n=1[infinity]an.